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THE  ELEMENTS 


OF 


HYDRAULIC  ENGINEERING 


PREPARED  FOR  STUDENTS  OF 

THE  INTERNATIONAL  CORRESPONDENCE  SCHOOLS 
SCRANTON,  PA. 


Volume  IV 


TABLES  AND   FORMULAS 


first  Edition 


SCRANTON 

THE  COLLIERY  ENGINEER  CO. 
1808 


441 


Entered  according  to  the  Act  of  Congress,  in  the  year  1898,  by  THE  COLLIERY 

ENGINEER  COMPANY,  in  the  office  of  the  Librarian  of  Congress, 

at  Washington. 


BURR  PRINTING  HOUSE, 
FRANKFORT  AND  JACOB  STREETS, 

NEW  YORK. 


1  2 

UNIVERSITY  OF  CALIFORNIA 
)  4-  SANTA  BARBARA  COLLEGE  LIBRARY 


TABLES   AND    FORMULAS. 


This  volume  contains  all  the  principal  Tables  and 
Formulas  which  are  likely  to  be  used  by  the  student  in 
practice.  They  have  been  collected  and  placed  in  this 
volume  in  order  to  make  them  convenient  for  ready  refer- 
ence, so  that  the  student  will  not  be  obliged  to  hunt  them 
out  in  the  preceding  volumes.  The  number  after  each 
formula  is  the  same  as  the  number  following  the  same 
formula  in  the  text.  '. 


4-4-3 


TABLK 

OP 

COMMON   LOGARITHMS 

OK    NUMBERS 

From    1    to  1  0.OOO. 

X.           Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

0 

-          00 

20 

3^  103 

40 

60  206 

60 

77  815 

80 

90  309 

i 

OO   OOO 

21 

32    222 

4i 

61  278 

61 

78  533 

81 

90  849 

2 

30    103 

22 

34  242 

42 

62  325 

62 

79  239 

82 

91  381 

3 

47  712 

23 

36  173 

43 

63  347 

63 

79  934 

83 

91  908 

4 

60  206 

24 

38   021 

44 

64  345 

64 

80  618 

84 

92  428 

5 

69  897 

25 

39  794 

45 

65  321 

65 

81  291 

85 

92  942 

6 

77  8i5 

26 

4i  497 

46 

66  276 

66 

81  954 

86 

93  45o 

7 

84  510 

27 

43  136 

47 

67    210 

67 

82  607 

8? 

93  952 

8 

90  309 

28 

44  7i6 

48 

68  124 

68 

83  251 

88 

94  448 

9 

95  424 

29 

46  240 

49 

69  020 

69 

83  885 

89 

94  939 

10 

00    000 

30 

47  712 

50 

69  897 

ro 

84  5io 

90 

95  424 

ii 

04  139 

3i 

49  136 

5i 

70  757 

71 

85  126 

91 

95  904 

12 

07  918 

32 

50  515 

52 

71  600 

72 

85  733 

92 

96  379 

13 

"  394 

33 

5i  851 

53 

72  428 

73 

86  332 

93 

96  848 

M 

14  613 

34 

53  148 

54 

73  239 

74 

86  923 

94 

97  313 

15 

17  609 

35 

54  407 

55 

74  036 

75 

87  506 

95 

97  772 

16 

20  412 

36 

55  630 

56 

74  819 

76 

88  081 

96 

98  227 

17 

23  045 

37 

56  820 

57 

75  587 

77 

88  649 

97 

98  677 

18 

25   527 

38 

57  978 

58 

76  343 

78 

89  209 

98 

99  "3 

19 

27  875 

39 

59  106 

59 

77  085 

79 

89  763 

99 

99  564 

20 

30  103 

40 

60  206 

60 

77  8i5 

80 

90  309 

100 

OO   OOO 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

p. 

100 

oo  ooo 

043 

087 

130 

173 

217 

260 

303 

346 

389 

101 

432 

475 

518 

56i 

604 

647 

689 

732 

775 

817 

102 

103 

104 

105 
1  06 
107 

108 
109 

860 
oi  284 

703 

02  Ilg 
531 
938 
03  342 

743 

9^3 
326 
745 
160 
572 
979 
383 
782 

94S 
368 
787 

202 

612 
*oig 
423 

822 

988 
410 
828 
243 
653 
*o6o 
463 
862 

*030 
452 
870 
284 
694 

*IOO 

503 
902 

494 
912 

325 
735 
*i4i 
543 
941 

*iiS 

536 
953 
366 
776 
*iSi 
583 
981 

AibV 
578 
995 
407 
816 

*222 

623 

*02I 

"199 
620 
*036 
449 

857 

*262 
663 

*o6o 

••242 
662 
*078 

49° 
898 

*302 

703 

*IOO 

i 

3 

4 

I 
7 
8 
9 

44 

4-4 

13.2 
17.6 

26'.4 
30.8 
35-2 
39-6 

43 

s'i 

12.9 
17.2 

25*8 

30.1 

34-4 
38.7 

42 

4- 
8. 

12. 

16. 

25- 
29. 
33-6 
37-8 

110 

04  139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

in 

112 
H3 
114 

H5 

116 
117 
118 
119 

532 
922 
05  308 
690 

06  070 
446 
819 
07  188 

555 

57i 
961 
346 
729 
108 
483 
856 
225 
59i 

610 
999 
385 
767 
145 
52i 
893 
262 
628 

650 
*o38 
423 
805 
183 
558 
930 
298 
664 

689 

*o?7 
461 
843 

221 

595 
967 
335 
700 

727 
*H5 
500 
881 
258 
633 
*oo4 
372 
737 

766 
*I54 
538 
918 
296 
670 
*o4i 
408 
773 

805 
*I92 
576 
956 

333 
707 
*07S 
445 
809 

844 

*23I 

614 

994 
371 

744 
*ii5 
482 
846 

883 

*269 

652 

*032 
408 
781 
*I51 

518 
882 

3 
4 

7 

9 

41 

4.1 

12.3 
16.4 

i:" 

IG'.O 

40 

12.0 

16.0 
24.0 
32.0 

39 

3-9 
7  -8 

15.6 
19-5 
23.4 
27.3 
31.2 
35-1 

120 

918 

954 

990 

*027 

*o63 

*°99 

*I35 

*i7i 

*207 

*243 

121 

122 
123 
124 
125 
126 
127 
128 
I29 

08  279 
636 
991 
09  342 
691 
10  037 
380 
721 
it  059 

314 

672 

*026 

377 
726 
072 
415 
755 
093 

350 
707 
*o6i 
412 
760 
106 
449 
789 
126 

386 

743 
*og6 
447 
795 
140 
483 
823 
1  60 

422 

778 

*I32 

482 
830 
175 
517 
857 
193 

458 
814 
*i6? 
517 
864 
209 
55i 
890 
227 

493 
849 

*202 

552 
899 
243 

585 

924 
261  J 

529 
884 
*237 
587 
934 
278 
619 
958 
294 

565 
92O 
*272 
621 
968 
3" 

11 

327 

600 
955 
*3Q7 
656 

*003 

346 
687 

*025 

361 

3 
4 
5 
6 
7 

9 

38 

3-8 
7.6 
11.4 
15-2 

22^8 
26.6 
3°-4 

34-2 

37 

3-7 
7-4 

18.5 

ag'6 
33-3 

36 

3.6 
7.2 

14.4 

21.6 

25.2 
28.8 
32-4 

130 

394 

428 

461 

494 

528 

56i 

594 

628 

661 

694 

131 
132 
133 
134 
135 
136 
137 
138 

727 
12  057 
385 
710 
13  033 
354 
672 
988 

760 
090 
418 
743 
066 
386 
704 
*oig 

793 
123 

45o 
775 
098 
418 
735 
*osi 

826 
156 
483 
808 
130 
450 
767 

*082 

860 
189 
5i6 
840 
162 
481 

*799 
*H4 

893 

222 
548 
872 
I94 
513 
830 

*I45 

926 
254 
58i 
905 
226 
545 
862 
*i76 

959 
287 
613 

937 

258 
577 
893 

*208 

992 
320 
646 
969 
290 
609 
925 
*239 

*024 

352 
678 

*OOI 

322 

640 

956 

*2?0 

2 

3 
4 
5 
6 

7 

35 

34 

3-4 
6.8 

13-6 
17.0 
20.4 
23.8 

33 

6.6 
9-9 
13.2 
16.5 
ig.8 
g.« 

139 

14  3oi 

333 

364 

395 

426 

457 

489 

520 

55i 

582 

9 

3I-S 

30.5 

29.7 

140 

613 

6«4 

675 

706 

737 

768 

799 

829 

860 

891 

141 
142 
143 
144 
145 
146 
147 

922 
15  229 
534 
836 
16  137 
435 
732 

953 
259 
564 
866 
167 
465 
761 

983 
290 
594 
897 
197 
495 
791 

*OI4 

320 
625 
927 

227 
524 
820 

*<H5 
35i 
655 
957 

256 
554 
850 

*076 
38i 
685 
987 
286 
584 
879 

*io6 
412 
715 
*oi7 
316 
613 
909 

*I37 
442 
746 
*047 
346 
643 
938 

*i68 
473 
776 

*o77 
376 
673 
967 

*igS 
503 
806 
*io7 
406 
702 
997 

2 

3 
4 
5 
6 

32 

i:: 

Q.6 
12.8 

19.2 

31 

1:1 

9-3 
1.1 

30 

6.0 

148 
149 

17  026 
319 

056 

348 

085 
377 

114 
406 

143 

435 

173 
464 

202 

493 

231 
522 

260 

55i 

289 
58o 

9 

i&i 

27.9 

*4-° 

150 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

p. 

P. 

LOGARITHMS. 


N. 

L.  o  |  i 

2 

3 

4 

5 

6 

7 
Sii 

8 

9 

P.  P. 

150 

J51 
152 
153 
154 
155 
156 
157 
158 
T59 
160 
161 
162 
163 
164 
165 
1  66 
167 
1  63 
169 

uo 

-i 

73 
74 
75 
76 
77 
78 
79 
180 
Si 
82 
183 
184 
185 
186 
187 
188 
189 

190 

191 
192 
IQ3 
194 
195 
196 
197 
198 
199 
200 

17  609 

638 

66? 

955 
241 
526 
808 
089 
368 
645 
921 
194 

696 

984 
270 
554 
837 
117 
396 
673 
948 

222 

725 

754 

782 

840 

869 

2 

3 
4 
5 
6 
7 
8 
9 

I 
9 

7 
8 
9 

3 

4 
5 
6 
7 
8 
9 

29 

I'.l 
8-7 
n.6 
'4-5 
17-4 
20.3 

ll.l 

S 

27 

2. 

5- 
8. 

i: 

£S 

24.3 

2 

2 

5 
;   7 

12 

15 

'7 

8   20 
9    22 

24 

2.4 
4.8 

7-2 

9.6 

12.0 
14.4 

16.8 

IT.l 

22 

2.2 

u 

8.8 
'3-2 

£1 

19.8 

28 

2.8 

5-6 
8. 

16. 

19. 

25- 

'& 
j.1 

II- 
23.4-' 

5 
23 

::i 

6.9 

9.2 

i! 

18.4 

ZQ.J 

21 

2.1 
4.2 
6.3 

8.4 

!°:I 
14.7 

16.8 
18.9 

898 
18  184 

469 
752 
19  033 
312 

590 

866 

20  140 

926 
213 
498 
780 
06  1 
340 
618 
893 
167 

*oi3 
298 
583 
865 
145 
424 
700 
976 
249 

*04I 

327 
611 
893 
173 
451 
728 
*oo3 
276 

*o7o 
355 
639 
921 

201 

479 
756 
*030 
303 

*o99 
384 
667 
949 
229 
507 
783 
*o58 
330 

*I27 
412 
696 

977 
257 
535 
811 
*o85 
358 

*i56 
441 

724 
*oo5 
285 
562 
838 

*II2 

385 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

683 
952 
21  2ig 
484 
748 
22  Oil 

272 
531 

789 

710 
978 
245 
5ii 
775 
037 
298 
557 
814 

737 
*oo5 

272 

537 
So  r 
063 
324 

583 
840 

763 

*032 
299 
564 

827 
089 
350 
608 

866 

790 
*059 

325 
59° 
854 
H5 
376 
634 
891 

817 
*oSs 
352 
617 
880 
141 
401 
660 
9i7 

844 

*II2 

378 
643 
906 
167 
427 

686 
943 

871 
*i39 
405 
669 
932 
194 
453 
712 
968 
223 

477 
729 
980 
229 
477 
724 
969 

212 

455 

308 
*i&5 
43i 
696 
958 
220 
479 
737 
994 

925 
*I92 

458 
722 
985 
246 

505 

763 

*oig 

23  045 

070 

096   121 

147 

172 

198 

249 
502 
754 
*o&5 
254 
502 
748 
993 
237 
479 

274 

528 

779 
*030 

279 
527 
773 
*oi8 
261 
503 

300 

553 
805 
24  055 
304 
55i 
797 
25  042 
285 

325 
578 
830 
080 
329 
576 
822 
066 
310 
55i 

350 
603 
855 
105 

353 
601 
846 
091 
334 

376 
629 
880 
130 
378 
625 
871 
iis 
358 

401 

654 
905 
155 
403 
650 
895 
139 
382 

426 
679 
930 
1  80 
428 
674 
920 
164 
406 

452 
704 
955 
204 
452 
699 
944 
188 
431 

527 

575 

600 
840 
079 

316 

553 
788 

*02I 
254 
485 
715 

624 

648 

672 

696 
935 
174 
411 

647 
881 
*ii4 
346 
577 
807 

*o35 

720 

959 
198 

435 
670 
905 
*i38 
370 
600 
830 
*os8 

~2^T 

5" 

735 
959 
181 

403 

623 
842 
*o6o 

744 

768 
26  007 
245 
482 
7i/ 
95i 
27  184 
416 
646 

792 

031 
269 
505 
74i 
975 
207 

439 

^L 

898 

~I26~ 

353 
578 
803 
026 
248 
469 
688 
J>07_ 
125 

816 

055 
293 
529 
764 
998 
231 
462 
692 

864 

102 
340 
576 

811 
*045 

277 
508 
738 

888 
126 
364 
600 
834 
*o6S 
300 
531 
761 

912 
150 

387 
623 

858 

*09I 

323 

554 
784 

983 

221 

458 
694 
928 

*i6i 
393 
623 
852 
*o8i 

875 

921 

944 

967 
194 
421 
646 

870 
092 
3r4 
535 
754 
_973_ 
190 

989 
217 
443 
668 
892 
"5 
336 
557 
776 
994 

211 

*OI2 

28  103 
330 
556 
780 
29  003 
226 
447 
667 
885 

149 

375 
601 
825 
048 
270 

49  1 
710 

929 

171 

398 
623 
847 
070 
292 
513 
732 
..9SL 
1  68 

240 
466 
69I 
914 
137 

358 
579 
798 
*oi6 

262 
488 
713' 
937 
159 
380 
601 
820 
*038 

307 
533 
758 
981 
203 
425 
645 
863 
*o8i 

30  103 

146 

233 

255 

276 

298 

N. 

L.  o 

I   j   2 

3 

4 

5  1  6 

7  |  8 

9 

P.  P. 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P. 

200 

30  103 

125 

146 

1  68 

190 

211 

233 

255 

276 

298 

201 
202 
203 
204 
205 
206 
207 
2O8 
209 

320 
535 
750 
963 
3i  175 
387 
597 
806 
32  015 

34i 
557 
771 
984 
197 
408 
618 
827 
035 

363 

5/8 
792 
*oo6 
218 
429 

639 
848 
056 

384 
600 
814 

*027 

239 
450 
660 

869 

077 

406 
621 
835 
*o48 
260 
471 
68  1 
890 
098 

42S 
643 
856 

*o6g 
281 
492 
702 
911 
118 

449 
664 
878 
*ogi 
302 
513 
723 
931 
139 

47i 

685 
899 

*II2 
323 

534 
744 
952 
160 

492 
707 
920 
*I33 
345 
555 
765 
973 
181 

5M 
728 
942 
*i54 
366 
576 
785 
994 

201 

3 

4 
5 
6 
7 
8 
9 

i 
i 

2    21 

.2    2.1 
•4     4-2 

.6   6/3 
.8   8.4 

:!  5:2 

).8   18.9 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

211 
212 
213 
214 
215 

216 

217 

218 

2I9 

428 
634 
838 

33  041 
244 
445 
646 
846 
34  044 

449 
654 
858 
062 
264 
465 
666 
866 
064 

469 
675 
879 
082 
284 
486 
686 
885 
084 

490 

695 
899 

102 
304 
506 
706 

9°5 
104 

5io 
715 
919 

122 
325 
526 
726 

925 
124 

531 
736 
940 
143 
345 
546 
746 
945 
143 

552 
756 
960 
163 
365 
566 
766 
965 
163 

572 
777 
980 
183 
385 
586 
786 
985 
183 

593 
797 

*OOI 

203 
405 
606 
806 
*oo5 
203 

6I3 

818 

*02I 
224 
425 
626 
826 
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223 

3 
4 

I 
9 

20 

e!o 

8.0 

14.0 
16.0 
18.0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

221 
222 
223 
224 
225 
226 
227 
228 
229 

439 
635 
830 
35  025 
218 
411 
603 
793 
984 

459 
655 
850 
044 
238 
430 
622 
813 
*oo3 

479 
674 
869 
064 
257 
449 
641 
832 

*02I 

498 
694 
889 
083 
276 
468 
660 
851 

*040 

518 
713 
908 
IO2 

295 

488 

679 

870 

*059 

537 
733 
928 

122 
315 
507 
698 
889 

*o?8 

557 
753 
947 
141 
334 
526 
717 
908 
*og7 

577 
772 
967 
160 
353 
545 
736 
927 
*ii6 

596 
792 
986 
1  80 
372 
564 
755 
946 
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616 
8n 
*oo5 
199 
392 
583 
774 
965 
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3 
4 

I 
9 

19 

•9 
.8 

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•5 
i  .4 
1  -3 

I  .2 

230 

36  J73 

192 

211 

229 

248 

267 

286 

305 

324 

342 

231 
232 
233 
234 
235 
236 
237 
238 
239 

361 
549 
736 
922 
37  107 
291 
475 
658 
840 

380 
568 
754 
940 
125 
310 
493 
676 
858 

399 

586 

773 
959 
144 
328 
5" 
694 
876 

418 
605 

791 

977 
162 
346 
530 
712 
894 

436 
624 
810 
996 
181 
365 
548 
73i 
912 

455 
642 
829 

*OI4 

199 
383 
566 
749 
93i 

474 
661 
847 
*o33 
218 
401 
585 
767 
949 

493 
680 
866 

*05I 

236 
420 
603 

785 
967 

5" 

698 
884 
*o7o 
254 
438 
621 
803 
985 

530 
717 
903 

*o8S 
273 
457 
639 
822 

*003 

3 
4 

i 
I 

9 

18 

.8 
.6 
•4 

.2 
12^6 

\l:l 

240 

38  021 

°39 

057 

075 

093 

112 

130 

148 

1  66 

184 

241 
242 
243 
244 
245 
246 
24? 
248 
249 

202 
382 
561 

739 
917 
39  094 
270 
445 
620 

220 

399 

578 
757 
934 
in 
287 
463 
637 

238 
417 
596 
775 
952 
129 
3°5 
480 
655 

256 
435 
614 
792 
970 
146 
322 
498 
672 

274 
453 
632 
810 
987 
164 
340 
515 
690 

292 
471 
650 

828 

*oo5 
182 
358 
533 
707 

3io 
489 
668 
846 

*023 

199 

375 
550 
724 

328 
507 
686 
863 
*04i 
217 
393 
568 
742 

346 

525 
703 
881 
^058 
235 
410 
585 
759 

364 

543 
721 
899 
*076 

252 
428 
602 

777 

i 

3 
4 

1 
7 
8 
9 

17 

'•7 
3-4 
5-' 
6.8 
••5 

s 

15-3 

250 

794 

8n 

829 

846 

863 

SSi 

898 

915 

933 

950 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

J 

P. 

p. 

LOGARITHMS. 


N. 

L.  o 

' 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

250 

39  794 

811 

829 

846 

863 

881 

898 

9i5 

933 

95° 

251 
252 
253 
254 
255 
256 
257 
258 
259 

967 
40  140 
312 
483 
•  654 
824 
993 
41  162 
330 

985 
157 
329 
500 
671 
841 

*OIO 

179 

347 

*002 
175 

346 

518 

688 

858 

*02? 

196 
363 

*oi9 
192 
364 

535 
705 
875 
*<M4 

212 
380 

*o37 
209 
38i 
552 
722 
892 

*06! 

229 

397 

*°54 
226 
398 
569 
739 
909 
*078 
246 
414 

*07I 

243 
4i5 
586 
756 
926 
*o95 
263 
430 

*o88 
261 
432 
603 
773 
943 
*ni 
280 
447 

*io6 

278 

449 
620 

79° 
960 

*I28 

296 

464 

*I23 

295 
466 
637 

807 

976 

*i45 
313 
481 

2 

3 
4 

I 
7 
8 
9 

18 

3-6 
5-4 
7.2 

Q 

ll'.6 
14.4 
16.2 

260 

497 

514 

531 

547 

564 

58i 

597 

614 

631 

647 

261 
262 
263 
264 
265 
266 
267 
268 
269 

664 
830 
996 
42  160 
325 
488 
651 
8i3 
975 

68  1 
847 

*OI2 
177 
341 
504 
667 
830 
991 

697 
863 

*02g 
193 
357 
52i 
684 
846 
*oo8 

7*4 

880 
*045 

2IO 

374 

537 
700 
862 

*024 

73i 
896 

*062 

226 
39° 

553 
716 
878 
*040 

747 
9T3 
*o78 
243 
406 
570 
732 
894 
*os6 

764 
929 
*o95 
259 
423 
586 

749 
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*0?2 

780 
946 
*m 
275 
439 
602 
765 
927 
*o88 

797 
963 

*I27 

292 

455 
619 
781 
943 
*I04 

814 

979 
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308 
472 
635 
797 
959 

*I20' 

3 
4 

'    I 

9 

17 

1-7 
3-4 

1:1 

ti'i 

i:S 

15-3 

270 

43  136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

271 

272 
273 
274 
275 
276 
277 
278 
279 

297 
457 
6x6 

775 
933 
44  091 
248 
404 
560 

313 

473 
632 
791 
949 
107 
264 
420 
576 

329 
489 
648 
807 
965 

122 
279 
436 

592 

345 
505 
664 
823 
981 
138 
295 
45i 
607 

361 

521 
680 

838 

996 
154 
311 

467 
623 

377 
537 
696 

854 

*OI2 
170 
326 

483 
638 

393 
553 
712 
870 

*028 

185 
342 
498 

654 

409 
569 
727 
886 
*044 
2QI 

358 
514 
669 

425 
584 
743 
902 

*059 
217 

373 
529 
685 

441 
600 

759 
917 

*Q75 
232 
389 
545 
700 

i 

2 

3 
4 

i 

7 
8 
9 

16 

1.6 

b 

6.4 

8.0 
9.6 

12^8 

14.4 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

281 
282 
283 
284 
285 
286 
287 
288 
289 

871 
45  025 
179 
332 
484 
637 
788 

939 
46  090 

886 
040 
194 
347 
500 
652 
803 
954 
105 

902 
056 
209 
362 
5i5 
667 
818 
969 

120 

917 
071 

225 
378 
530 
682 

834 
984 
135 

932 
086 
240 

393 
545 
697 
849 

*000 

150 

948 

102 
255 
408 
561 
712 
864 

*ois 
165 

963 
117 
271 
423 
576 
728 

879 
*030 
1  80 

979 
133 
286 

439 
59i 
743 
894 

*045 
195 

994 
148 
301 
454 
606 
758 
909 
*o6o 

210 

*OIO 

163 
317 
469 
621 
773 
924 
*075 
225 

2 

3 
4 

I 
7 
8 
9 

15 

3- 

4-   y 

7- 
9- 

!3- 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

' 

291 
292 
293 
294 
295 
296 
297 
298 
299 

389 
538 
687 
835 
982 
47  129 
276 
422 
567 

404 
553 
702 
850 
997 
144 
290 
436 
582 

419 
568 
716 
864 
*OI2 
159 
305 
451 
596 

434 

583 
731 
879 

*026 

173 
319 

465 

611 

449 
598 
746 
894 

*04I 

188 
334 
480 
625 

464 
613 
761 
909 
*Q56 

202 

349 
494 
640 

479 
627 
776 
923 

*070 

217 

363 

509 

654 

494 
642 
790 
938 
*o85 
232 
378 
524 
669 

509 
657 
805 
953 

*IOO 

246 
392 

538 
683 

523 
672 
820 
967 
*H4 
261 
407 
553 
698 

3 

4 

I 
7 
8 
9 

14 

1.4 

2.8 

5-6 

7.0 

b 

II.  2 
12.6 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

300 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

301 
302 
303 
304 
305 
306 
307 
308 
309 

857 
48  ooi 
144 
287 
430 
572 
7M 
855 
996 

871 
015 

159 
302 

444 

586 
728 
869 

*OIO 

885 
029 
173 
316 

458 

601 

742 

883 

*024 

900 

044 
187 
330 
473 
615 
756 
897 
*o38 

914 

058 

202 

344 
487 
629 
770 
911 

*052 

929 
0/3 
216 
359 
5oi 
643 
785 
926 
•*o66 

943 
087 
230 
373 
515 
657 
799 
940 
*oSo 

958 

101 

244 

387 

530 
671 
813 

954 
*o94 

972 
116 

259 
401 

544 
686 
827 
968 
*io8 

986 
130 
273 
416 
558 
700 
841 
982 

*I22 

2 

3 
4 

15 

i  -5 

4- 

7. 
9- 

310 

49  136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

7 

10. 

3" 

312 
313 
314 
315 
316 
317 
3i8 
319 

276 
415 
554 
693 
831 
969 
50  1  06 
243 
379 

290 
429 

568 
707 

845 
982 

120 

256 
393 

304 

443 

582 
721 
85<J 
996 
133 
270 
406 

3i8 
457 
596 
734 

872 

*OIO 

147 

284 
420 

332 
471 

610 

748 
886 

*024 

161 

297 
433 

346 
485 
624 
762 
900 
*037 
i?4 
3" 
447 

360 

499 

638 
776 
914 

*05I 

188 
325 
461 

374 
513 
651 
790 
927 
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202 
338 
474 

388 
527 
665 
803 
941 
*079 
215 
352 
488 

402 
541 
679 
8l7 

955 

*092 

229 
365 
5oi 

9 

13- 
14 

2.8 

4.2 

5-6 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

l:\ 

321 
322 
323 
324 
325 
326 
327 
328 
329 

651 

786 
920 
5i  055 
188 
322 
455 
587 
720 

664 
799 

934 
068 

2O2 

335 
468 
601 
733 

678 
813 
947 
08  1 
215 
348 
481 
614 
746 

691 
826 
961 
095 
228 
362 
495 
627 
759 

705 
840 
974 
108 
242 
375 
508 
640 
772 

718 
853 
987 

121 

255 
388 
521 

654 

786 

732 
866 

*OOI 

135 
268 
402 
534 
667 
799 

745 
880 

*OI4 

148 
282 
415 
548 
680 
812 

759 

S<J3 

*028 

162 
295 

428 
561 
693 

825 

772 
907 

*04I 

175 

308 

441 

574 
706 

838 

9 

2 

3 

Q.8 
II.  2 
12.6 

13 

I'l 

3-9 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

4 

33i 

332 
333 
334 
335 
336 
337 
338 
339 

983 
52  114 
244 
375 
504 
634 
763 
892 
53  020 

996 
127 
257 
388 
5W 
647 
776 
905 
033 

^009 
140 
270 
401 
530 
660 
789 
917 
046 

*022 

153 
284 

414 

543 
673 
802 
930 
058 

*o35 
1  66 
297 
427 
556 
686 
8i5 
943 
071 

*048 
179 
310 
440 
569 
699 
.827 
956 
084 

*o6i 
192 
323 
453 
582 
711 
840 
969 
097 

*075 
205 
336 
466 
595 
724 
853 
982 
no 

*o88 
218 
349 
479 
608 

737 
866 
994 

122 

*IOI 

231 
362 
492 
621 
750 
879 
*oo7 

135 

6 
7 
8 
9 

\'l 

Q.I 
II.7 

12 

1.2 

340 

148 

161 

173 

1  86 

199 

212 

224 

237 

250 

263 

3 

3-6 

34i 
342 
343 
344 
345 
346 
347 
348 
349 

275 
403 
529 
656 
782 
908 
54  033 
158 
283 

288 
415 
542 
668 
794 
920 
045 
170 
295 

301 

428 
555 
681 
807 
933 
058 
183 
307 

3U 
441 
567 
694 
820 
945 
070 
195 
320 

326 
453 
580 
706 
832 
958 
083 
208 
332 

339 
466 

593 
719 
845 
970 
095 

220 

345 

352 
479 
605 
732 
857 
983 
108 
233 
357 

364 

491 
618 
744 
870 
995 

120 

245 

370 

377 
504 
631 

757 
882 
*oo8 
133 
258 
382 

39° 
517 
643 
769 
895 

*020 
145 
270 

394 

4 
5 
6 
7 
8 
9 

4.8 

K 

9.6 

350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

5i8 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P 

p. 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

350 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

35i 
352 
353 
354 
355 
356 
357 
358 
359 

531 
654 
111 
900 
55  023 
M5 
267 
388 
509 

543 
667 
790 
913 
035 
157 
279 
400 
522 

555 
679 
802 
925 
047 
169 
291 
413 
534 

568 

691 
814 
937 
060 
182 
303 
425 
546 

580 
704 

827 

949 
072 
194 

315 
437 

558 

593 
716 

839 
962 
084 
206 
328 
449 
570 

605 
728 
851 
974 
096 
218 
340 
461 
582 

617 
741 
864 
986 
108 
230 
352 
473 
594 

630 
753 
876 
998 

121 
242 
364 
485 
606 

888 
*on 
133 
255 
376 
497 
618 

2 

3 
4 
5 
6 
7 

13 

1:1 

'3-q 

5 

9.1 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

8 

10.4 

361 
362 
363 
364 
365 
366 
367 
368 
369 

871 
991 
56  no 
229 

348 
467 
585 
703 

763 
^883 

122 
241 
360 

478 

597 
714 

775 
895 
*oi5 
134 
253 
372 
490 
608 
726 

787 

*02'7 

146 

265 
384 

502 

620 

738 

799 

158 
277 
396 

750 

811 

*oso 
170 
289 
407 
526 
644 
761 

823 
943 

*062 

182 
301 
419 

538 
656 
773 

835 

*o55 
194 
312 

549 
667 

785 

847 
967 

*o86 
205 
324 
443 
56i 
679 
797 

859 
979 
*og8 
217 
336 
455 
573 
691 
808 

2 

3 
4 

J 

12 

4-8 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

| 

7.2 

372 
373 
374 
375 
376 
377 
378 
379 

937 

57  054 
171 
287 
403 
519 
634 
749 
864 

949 
066 
183 
299 
415 
530 
646 
761 
875 

961 
078 
194 
310 
426 
542 
657 
772 
887 

972 
089 
206 
322 

438 

553 
669 
784 
898 

984 

IOI 

217 

334 
449 
565 
680 
795 
910 

996 

229 
345 
461 
576 
692 
807 
921 

*oo8 
124 
241 

-357 
473 
588 
703 
818 
933 

*oig 
136 
252 
368 
484 
600 
715 
830 
944 

*03I 

148 
264 
380 
496 
611 
726 
841 
955 

159 
276 
392 
507 
623 
738 
852 
967 

7 
8 
9 

8 

10.8 

380 

978 

990 

*OOI 

*oi3 

*024 

*o35 

*047 

*os8 

•081 

3 

3-3 

382 
383 
384 
385 
386 
387 
388 
389 

58  092 
206 
320 
433 
546 
659 
771 
883 
995 

104 

213 

33i 
444 
557 
670 
782 
894 
*oo6 

"5 
229 
343 
456 
569 
68  1 
794 
906 
*oi7 

127 
240 
354 
467 
580 
692 
805 
917 

*028 

138 

252 
365 
478 

591 
704 
816 
928 

*040 

149 
263 
377 
490 
602 
715 
827 

*939 

161 

274 
388 
SGI 
614 
726 
838 
950 

*062 

172 

286 

399 
512 
625 
737 
850 
961 
*073 

184 
297 
410 
524 
636 

749 
861 

*973 

195 

309 
422 
535 
647 
760 
872 
984 
*o95 

1 

I 
g 

1:1 

9-9 

10 

390 

59  Io6 

118 

129 

140 

151 

162 

173 

184 

195 

207 

i 

1.0 

392 
393 
394 
395 
396 
397 
398 
399 

21  8 

329 
439 
550 
660 
770 
879 
988 
60  097 

229 
340 
450 

671 
780 
890 

999 
1  08 

240 

461 
572 
682 
791 

*OIO 

119 

251 

362 

47-' 
583 
693 
802 
912 

*02I 
130 

262 
373 
483 
594 
704 
813 
923 

*032 
141 

273 
384 
494 
605 
715 
824 
934 
*o43 
152 

284 

395 
506 
616 
726 
835 
945 
*054 
163 

295 
406 
517 
627 
737 
846 
956 
*o65 
173 

306 
417 

528 
638 
748. 
857 
966 

184 

428 

539 
649 

759 
868 
977 
*o86 
195 

3 
4 

1 
7 
8 
9 

400 

206 

217 

228 

239 

240 

260 

271 

282 

293 

3°4 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

9 

P. 

P. 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 
282 

8 

9 

P.  P. 

400 

60  206 

217 

325 
433 
54i 
649 
756 
863 
970 
077 
183 

228 

~33&" 

444 
552 
660 
767 
874 
981 
087 
194 

239 

249 

260 

271 

293 

304 

i 

3 
4 

1 

9 

3 
4 

i 

7 
8 
9 

2 

3 
4 

1 

9 

11 

6.6 

u 

9.9 

10 

3.0' 
I'o 

ti 

9 

i'.3 
2-7 
3-6 
4- 

7- 

401 
402 
403 
404 
405 
406 
407 
408 
409 
410 
411 
412 
413 
414 
415 
416 
417 
418 
419 
420 
421 
422 
423 
424 
425 
426 
427 
428 
429 
430 

43i 
432 
433 
434 
435 
436 
437 
438 
439 
440 
441 
442 
443 
444 
445 
446 
447 
448 
449 
450 

314 
423 
531 
638 
746 
853 
959 
61  066 
172 

347 
455 
563 
670 
778 
885 
991 
098 
204 

358 
466 
574 
68  1 

788 
895 

*002 
I0g 
215 

369 

477 
584 
692 

799 
906 
*oi3 
119 
225 

379 
487 
595 
703 
810 
917 

*023 

130 

236 

390 
498 
606 
7i3 
821 
927 
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140 
247 

401 
509 
617 
724 
831 
938 
*045 
151 
257 
363 
469 
574 
679 
784 
888 

993 
097 

201 
304 

412 
520 
627 

735 
842 
949 
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162 
268 

374 
479 
584 
690 
794 
899 
-003 
107 

211 
315 

4I8 

278 

289 

300 

3io 

321 
426 
532 
637 
742 
847 
951 
055 
159 
263 

33i 

342 

352 
458 
563 
669 
773 
878 
982 
086 
190 
294 

384 
490 
595 
700 
805 
909 
62  014 
118 

221 

395 
500 
606 
711 
815 
920 
024 
128 
232 
335 

405 
5" 
616 
721 
826 
930 
034 
138 
242 

416 
521 
627 
731 
836 
941 
045 
149 
252 
356 

437 
542 
648 
752 
857 
962 
066 
170 
273 

448 
553 
658 
763 
868 
972 
076 
1  80 
284 

325 

346 

366 

377 

387 

397 

408 

428 
531 
634 

737 
839 
941 
63  043 
144 
246 

439 
542 
644 
747 
849 
951 
053 
155 
256 

449 
552 
655 
757 
859 
961 
063 
165 
266 

459 
562 
665 
767 
870 
972 
073 
175 
276 

469 

572 
675 
778 
880 
982 
083 
185 
286 

480 
583 
685 
788 
890 
992 
094 
195 
296 

490 
593 
696 
798 
900 

*002 
104 
205 
306 

500 
603 
706 
808 
910 

*OI2 
114 
215 
317 

5" 
613 
716 
818 
921 

*022 
124 
225 

327 

521 
624 
726 
829 
931 

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134 
236 
J37. 
438 

347 

357 

367 

377 

387 
488 
589 
689 

789 
889 
988 
088 
187 
286 

397 

407 
508 
609 
709 
809 
909 

*oo8 
108 
207 
306 

417 

428 

448 
548 
649 

749 
849 

949 
64  048 
147 
246 

458 
558 
659 
759 
859 
959 
058 

157 
256 

355 
454 
552 
650 
748 
846 
943 
040 
137 
234 

468 
568 
669 
769 
869 
969 
068 
167 
266 

478 
579 
679 

779 
879 

979 

078 

177 
276 

498 

599 
699 

799 
899 
998 
098 
197 
296 

518 
6lg 
719 
8lg 
919 

*oi8 
118 
217 
316 

528 
629 
729 
829 

929 
*028 

128 
227 
326 

538 
639 
739 
839 
939 
*o38 
137 
237 
335 
434 
532 
631 
729 
826 
924 

*02I 

118 
215 
312 

345 

365 
464 
562 
660 
758 
856 
953 
050 
147 
244 

375 
473 
572 
670 
768 
865 
963 
060 
157 
254 

385 

_39S_ 
493 
59i 
689 

787 
885 
982 
079 
176 
273 

404 

4^4 

513 
611 
709 
807 
904 

*002 
Ogg 
I96 
292 

424 

444 
542 
640 
738 
836 
933 
65  031 
128 
225 

483 
582 
680 

777 
875 
972 
070 
167 
263 

503 
601 
699 
797 
895 
992 
089 
1  86 
283 

523 
621 

719 
816 
914 

*OII 

108 
205 

302 

321 

33i 

34i 

350 

360 

369 

379 

389 

398 

408 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

450 

45i 
452 
453 
454 
455 
456 
457 
458 
459 
460 
461 
462 
463 
464 
465 
466 
467 
468 
469 

4ro 

471 
472 

473 
474 
475 
476 
477 
478 
479 
480 
481 
482 
483 
484 
485 
486 
487 
488 
489 
490 
491 
492 
493 
494 
495 
496 
497 
498 
499 
500 

65  321 

331 

341 

35Q 
447 
543 
639 
734 
830 
925 

*020 

"5 

2IO 

360 

369 

379 
475 
57i 
667 
763 
858 

954 
*049 

143 
238 

_3S9_ 

485 
58i 
677 
772 
868 
963 
*o58 
153 
247 

^98_ 
495 
59i 
686 
782 
877 
973 
*o68 
162 
257 

408 

I 
g 

2 

3 
4 

i 

7 

8 
9 

3 
4 
5 
6 

I 
9 

10 

|;° 

l°0 

9.0 
9 

13 

2.7 

3-6 
4-5 
5-4 
6-3 
7.2 

8 

1.6 
2.4 
3.2 

4!s 

S-6 
6.4 
7.2 

418 
514 
610 
706 
Soi 
896 
992 
66  087 
181 

427 
523 
619 
715 

8n 
906 

*OOI 

096 
191 

285 

437 
533 
629 
725 
820 
916 

*OII 

106 

200 
295 

456 

552 
648 

744 
839 
935 
*O3O 
124 
219 
314 
408 
502 

783 
876 
969 
062 
154 

466 
562 
658 
753 
849 
944 
*039 
134 
229 

323 
417 
5" 
605 
699 
792 
885 
978 
071 
164 

504 
600 
696 
792 
887 
982 
*077 
172 
266 

276 

304 

332 

342 

351 

361 

370 
464 
558 
652 
745 
839 
932 
67  025 
H7 

380 
474 
567 
66  1 

755 
848 
941 
034 
127 

389 
483 

577 
671 
764 
857 
950 
043 
136 
228 
321 
413 
504 
596 
688 

779 
870 
961 
052 

398 
492 
586 
680 

773 
867 
960 
052 
145 
237 

427 
521 
614 
708 
801 
894 
987 
080 
173 

436 
530 
624 
717 
811 
904 
997 
089 
182 

445 
539 
633 
727 
820 

•$ 

099 
191 

455 
549 
642 
736 
829 
922 
*ois 
108 

2OI 

210 

219 

247 

256 

265 

274 

284 

293 

302 

394 
486 
578 
669 
761 
852 
943 
63  034 

3ii 
403 

495 
587 
679 
770 
861 
952 
043 

330 
422 
5U 
605 
697 
788 
879 
970 
06  1 

339 
431 
523 
614 
706 
797 
888 

979 
070 

348 
440 
532 
624 
715 
806 
897 
988 
079 
169 
260 
350 
440 
529 
619 
708 
797 
886 
975 
064 

357 
449 
54i 
633 
724 

8i5 
906 
997 

088 

367 
459 
550 
642 
733 
825 
916 
*oo6 
097 

376 
468 
560 
651 
742 
834 
925 
*oi5 
106 

385 

477 
569 
660 
752 
843 
934 

*024 

"5 

124 

133 

142 

151 

160 

251 
341 
431 
520 
610 
699 
789 
878 

966 
055 

178 

187 

196 
287 
377 
467 
556 
646 
735 
824 
9!3 

*002 
090 

205 

215 
305 
395 
485 
574 
664 
753 
842 
93i 

224 
314 
404 

494 
583 
673 
762 
851 
940 

233 
323 
413 
502 
592 
68  1 
771 
860 
949 

242 
332 
422 
5" 
601 
690 
780 
869 
958 

269 
359 
449 

538 
628 
717 
806 
895 
984 

278 
368 
458 
547 
637 
726 
815 
904 
993 

296 
386 
476 
565 
655 
744 
833 
922 

*OII 

099 

69  020 

028 

037 

046 

073 

082 

108 

197 

285 

373 
461 
548 
636 
723 
810 

117 
205 

3! 

469 

557 
644 
732 
819 

126 
214 
302 
390 
478 
566 
653 
740 
827 

135 
223 
3" 
399 

487 
574 
662 

749 

836 

144 

232 
320 
408 
496 
583 
671 
758 
845 
932 

152 
241 
329 
417 

504 
592 
679 
767 
854 
940 

161 
249 

338 
425 
513 
601 

688 
775 
862 

949 

170 

258 
346 
434 
522 
609 
697 
784 
871 
958 

179 
267 

355 
443 
531 
618 

705 
793 
880 

188 
276 
364 
452 
539 
627 
7M 
801 
888 

8Q7 

906  914 

923 

966 

975 

N. 

L.  o 

I 

2 

3 

4 

s 

6 

7 

8 

9 

P.  P. 

10 


LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

501 
502 
503 
504 
505 
506 
507 
508 
509 

984 
70  070 
157 
243 
329 
415 
5oi 
586 
672 

992 
079 
165 
252 
338 
424 
509 
595 
680 

*OOI 

088 
174 
260 

346 

432 

518 
603 

689 

*OIO 

096 
183 
269 

355 
441 
526 
612 
697 

*oi8 
105 
191 

278 
364 
449 
535 
621 
706 

*027 

114 

2OO 
286 
372 

458 

544 
629 
714 

*03& 

122 

209 

295 
38r 
467 
552 
638 
723 

*<H4 
131 
217 

303 
389 
475 
56i 
646 
73i 

*053 
140 
226 
312 
398 
484 
569 
655 
740 

*062 

148 
234 
321 
406 
492 
578 
663 
749 

3 

9 

o.g 
1.8 
2.7 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

4 

3.6 

5" 
512 
513 
514 
515 
5i6 
5i7 
518 
519 

842 
927 

71  012 
096 

181 

265 

349 
433 
517 

851 
935 
020 
105 
189 
273 
357 
441 
525 

859 
944 
029 
H3 
198 
282 
366 
450 
533 

868 
952 
037 

122 
206 
290 

374 
458 
542 

876 
961 
046 
130 
214 
299 
383 
466 
550 

885 
969 
054 
139 
223 
307 
39i 
475 
559 

893 
978 
063 
M7 
231 
315 
399 
483 
567 

902 
986 
071 

155 
240 
324 
408 
492 
575" 

910 
995 
079 
164 
248 
332 
416 
500 
584 

919 

*003 

088 
172 
257 
34i 
425 
508 
592 

6 
7 
8 
9 

7- 
8. 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

521 
522 
523 
524 
525 
526 
527 
528 
529 

684 
767 
850 
933 
72  016 
099 
181 
263 
346 

692 
775 
858 
941 
024 
107 
189 
272 
354 

700 

784 
867 
950 
032 
H5 
198 
280 
362 

709 
792 
875 
958 
041 
123 
206 
288 
370 

717 
800 
883 
966 
049 
132 
214 
296 
378 

725 
809 
892 
975 
057 
140 

222 
304 

387 

734 
817 
900 
983 
066 
148 
230 
3*3 
395 

742 
825 
908 
991 
074 
156 
239 
321 
403 

750 
834 
917 

999 
082 
165 

247 

329 
411 

759 
842 
925 
*oo8 
090 
173 
255 
337 
419 

i 
a 
3 
4 

I 

I 
9 

8 

'.6 
•  4_ 

5^6  ' 
6.4 
7.2 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

SGI 

53i 
532 
533 
534 
535 
536 
537 
533 
539 

509 
591 
673 
754 
835 
916 
997 
73  078 
159 

5i8 
599 
681 
762 
843 
925 
*oo6 
086 
167 

526 
607 
689 
770 
852 
933 
*oi4 
094 
175 

534 
616 
697 
779 
860 
941 

*022 
I  O2 
183 

542 
624 
705 
787 
868 

949 
*030 
in 

IQI 

550 
632 
713 

795 
876 
,957 
*038 
119 
199 

558 
640 
722 
803 
884 
965 
*046 
127 
207 

567 
648 
730 
811 
892 
973 
*054 
135 
215 

575 
656 
738 
819 
900 
981 

*062 

143 
223 

583 
665 
746 
827 
908 
989 
*o7o 
151 
231 

3 

7 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

4 

54i 

542 
543 
544 
545 
546 
547 
548 
549 

320 
400 
480 
56o 
640 
719 

799 

878 

957 

328 
408 
488 
568 
648 
727 
807 
886 
965 

336 
416 
496 
576 
656 
735 
815 
894 
973 

344 
424 
504 
584 
664 

743 
823 
902 
981 

352 
432 
512 
592 
672 
751 
830 
910 

989 

360 
44° 
520 
600 
679 
759 
838 
918 
997 

368 
448 
528 
608 
687 
767 
846 
926 
*oos 

376 
456 
536 
616 
695 
775 
854 
933 
*oi3 

384 

464 
544 
624 
703 
783 
862 
941 

*020 

392 
472 
552 
632 
711 
791 
870 
949 

*028 

6 

I 
9 

:1 

6.3 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

p. 

LOGARITHMS. 


11 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

55i 
552 
553 
554 
555 
556 
557 
553 
559 

H5 
194 
273 
351 
429 
507 
586 
663 
74  1 

123 

202 

280 

359 
437 
515 
593 
6/1 
749 

131 
2IO 

288 
367 

445 
523 
601 
679 

757 

139 
218 
296 
374 
453 
53i 
609 
687 
764 

i47 

225 
304 
382 
461 

539 
617 

695 

772 

155 
233 
312 

390 

468 

547 
624 
702 
780 

162 
241 

320 

398 
476 
554 
632 
710 
788 

170 
249 
327 
406 
484 
562 
640 
718 
796 

178 
257 
335 
414 
492 
570 
648 
726 
803 

186 

265 
343 
421 
500 

578 
656 

733 
811 

• 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

g  • 

561 
562 
563 
564 
565 
566 
567 
568 
569 

896 
974 
75  051 

128 
205 
282 
358 
435 
Sir 

904 
981 

059 
136 
213 
289 
366 
442 
519 

912 

989 
066 
143 
220 
297 
374 
450 
526 

920 
997 
074 
151 

228 
3°5 
38i 
458 
534 

927 
*oos 
082 
159 
236 
312 
389 
465 
542 

935 

*OI2 
089 

1  66 
243 
320 

397 
473 
549 

*943 

*O2O 
097 
174 
251 
328 
404 
48l 

557 

95o 

*028 

105 
182 
259 

335 
412 
488 
565 

958 
*o35 

"3 
189 
266 
343 
420 
496 
572 

966 
*043 

120 
197 
274 

351 
427 
504 
580 

i   0.8 

2    1.6 

3   2.4 
4   3-2 
5   4-° 
6   4.8 
7   5-6 
8   6.4 
9   7-2 

5?0 

587 

595 

603 

6ro 

618 

626 

633 

641 

648 

656 

57i 
572 
573 
574 
575 
576 
577 
573 
579 

664 
740 

8i5 

967 
76  042 
118 
193 

268 

671 
747 
823 
899 
974 
050 
125 
200 
275 

679 
755 
831 
906 
982 
057 
133 
208 
283 

686 
762 
838 
914 
989 
065 
140 
215 
290 

694 
770 
846 
921 
997 
072 
148 
223 
298 

702 
778 
853 
929 
*oos 
080 
155 
230 
305 

% 

861 
937 

*OI2 

087 
163 
238 
313 

717 
793 
868 
944 

*020 

095 
170 
245 
320 

724 
800 
876 
952 

*027 

103 

178 
253 

328 

732 
808 
884 

959 
*035 
no 
185 
260 
335 

580 

343 

350 

35? 

365 

373 

380 

388 

395 

403 

410 

58i 

582 
583 
584 
585 
586 
587 
588 
589 

418 
492 
567 
641 
716 
790 
864 
938 
77  012 

425 
500 
574 
649 
723 
797 
871 
945 
019 

433 
507 
582 
656 
730 
805 
879 
953 
026 

440 
515 
589 
664 
738 
812 
886 
960 
034 

448 
522 
597 
671 
745 
819 
893 
967 
041 

455 
530 
604 
678 
753 
827 
901 
975 
048 

462 

& 

686 
760 

834 
908 
982 
056 

470 

619 
693 
768 
842 
916 
989 
063 

477 
552 
626 
701 
775 
849 
923 
997 
070 

485 
559 
634 
708 
782 
856 
930 

*004 

078 

r 

I    -7 
2     .4 

4    !8 
1    * 

I    'I 

590 

085 

"93 

100 

107 

"5 

122 

129 

137 

144 

151 

I  1:5 

59i 
592 
593 
594 
595 
596 
597 
=98 
599 

159 
232 
305 
379 
452 
525 
597 
670 
743 

1  66 
240 
313 

386 

459 
532 
605 
677 
750 

173 
247 
320 
393 
466 

539 
612 
685 

757 

181 
254 
327 
401 
474 
546 
619 
692 
764 

188 
262 
335 
408 
481 
554 
627 
699 
772 

195 
269 
342 
415 
488 
561 
634 
706 

779 

203 
276 

349 
422 

495 
568 
641 

714 

786 

210 
283 

357 
430 
503 
576 
648 
721 
793 

217 
291 
364 
437 
5io 
583 
656 
728 
80  1 

225 
298 
371 
444 
517 
590 
663 
735 
808 

600 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

601 
602 
603 
604 
605 
606 
607 
608 
609 

887 
960 
78  032 
104 
176 
247 
3i9 
390 
462 

895 
967 

039 
in 
183 
254 
326 
398 
469 

902 

974 
046 
118 
190 
262 
333 
405 
476 

909 
981 
053 
125 
i97 
269 
340 
412 
483 

916 
988 
061 
132 
204 
276 
347 
419 
490 

924 
996 
068 
140 

211 

283 

355 
426 
497 

931 
*oo3 
075 
147 
219 
290 
362 
433 
504 

938 

*OIO 

082 
154 
226 
297 

369 
440 
512 

945 
*oi7 
089 
161 
233 
305 
376 
447 
519 

952 

*025 

097 
168 
240 
312 

383 
455 
526 

3 
4 

8 

1.6 
2.4 

3-2 

610 

533 

540 

547 

554 

56i 

569 

576 

583 

59° 

597 

S 
6 

4  8 

611 
612 
613 
614 
615 
616 
617 
618 
619 

604 
675 
746 
817 
888 
958 
79  029 
099 
169 

611 

682 
753 
824 
895 
965 
036 
106 
176 

618 
689 
760 
831 
902 
972 
043 
H3 
183 

625 

$ 

838 
909 

979 
050 

120 

190 

633 
704 
774 
845 
916 
986 
057 
127 
197 

640 
711 
78i 
852 
923 
993 
064 
134 
204 

647 
718 
789 
859 
930 

*000 

071 

141 

211 

654 
725 
796 
866 
937 
*oo7 
078 
148 
218 

66  1 
732 
803 
873 
944 
*oi4 
085 
155 
225 

668 

739 
810 
880 
95i 

*02I 
092 
162 
232 

I 
9 

5-6 
6.4 
7.2 

620 

239 

246 

253 

260 

267 

274 

28l 

288 

295 

302 

621 
622 
623 
624 
625 
626 
627 
628 
629 

309 

379 
449 
5i8 

588 
657 
727 
796 
865 

316 

386 
456 
525' 
595 
664 
734 
803 
872 

323 
393 
463 
S32 
602 
671 
74i 
810 
879 

330 
400 
470 

529 
609 
678 
748 
817 
886 

337 
407 
477 
546 
616 
685 
754 
824 
893 

344 
414 

484 
553 
623 
692 
761 
831 
900 

351 
421 
491 
560 
630 
699 
768 
837 
906 

358 
428 
498 
567 
637 
706 
775 
844 
913 

365 
435 
505 
574 
644 
713 
782 
851 
920 

372 

442 

5" 

58i 
650 
720 
789 
858 
927 

I 
2 

3 
4 

i 

7 
8 
9 

7 

1.4 

2.1 

3-5 
4.2 
4-9 
5-6 
6-3 

630 

934 

941 

948 

955 

962 

969 

975 

982 

983 

996 

631 
632 
633 
634 
635 
636 
637 
638 
639 

80  003 
072 
140 
209 
277 
346 
414 
482 
550 

OIO 

079 

147 
216 
284 
353 
421 
489 
557 

017 

085 
154 
223 
291 

359 
428 
496 
564 

024 
092 
161 
229 
298 
366 
434 
502 
570 

030 
099 
1  68 
236 
305 
373 
441 
509 
577 

037 
106 
r?5 
243 
312 
380 
448 
5i6 
584 

044 
U3 
182 
250 
3i8 
387 
455 
523 
59i 

051 

120 

188 
257 
325 
393 
462 
530 
598 

058 
127 
195 
264 
332 
400 
468 

536 
604 

065 
134 

202 
271 

339 
407 

475 
543 
611 

6 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

2.4 

641 
642 
643 
644 
645 
646 
647 
648 
649 

686 

754 
821 
889 
956 
81  023 
090 
158 
224 

693 
760 
828 
895 
963 
030 
097 
164 
231 

699 
767 
835 
902 
969 
037 
104 
171 
238 

706 
774 
841 
909 
976 
043 
in 
178 
245 

713 
781 
848 
916 
983 
050 
117 
184 
251 

720 
787 
855 
922 
990 
057 
124 
191 
258 

726 
794 
862 
929 
996 
064 
131 
198 
265 

733 
80  1 
868 
936 
*oo3 
070 
137 
204 
271 

740 
808 
875 
943 

*OIO 

077 
144 

211 

278 

747 
814 
882 

949 
*oi7 
084 
151 
218 
285 

7 
8 
9 

33:6 
4.2 
4.8 

5-4 

650 

291 

298 

305 

3" 

3i8 

325 

33i 

338 

345 

35i 

N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P. 

LOGARITHMS. 


13 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

650 

651 
652 
653 
654 
655 
656 
657 
658 

659 
660 

661 
662 
663 

664 
665 
666 
667 
668 
669 

6TO 

671 
672 
673 
674 
675 
676 
677 
678 
679 
680 
681 
682 
683 
684 
685 
686 
687 
688 
689 

690 

691 
692 
693 
694 
695 
696 
697 
698 
699 

roo 

81  291 

298 

305 

3" 

318 

325 

331 

338 

345 

35i 

r 

1-4 

2.1 

2.8 

3-5 
4.2 
4-9 
5-6 
6.3 

6 

i   0.6 

2   1.2 

3   1.8 
4   2-4. 

1  r.6 
7   4-2 
8   4.8 
9   5-4- 

358 
425 
491 
558 
624 
690 
757 
823 
889 

365 
43i 
498 
564 
631 
697 
763 
829 
895 

371 
438 
505 
571 
637 
704 
770 
836 
902 

378 
445 
5" 
578 
644 
710 
776 
842 
908 

385 
451 
5i8 
584 
651 
717 
783 
849 
_9i5_ 
981 

391 

458 

525 
591 
657 
723 
790 
856 

_9£1 

987 

053 
119 

184 

249 
315 

380 

445 
5io 
575 
640 

398 
465 
53i 
598 
664 
73o 
796 
862 
928 

405 
47i 
538 
604 
671 
737 
803 
869 
935 

411 
478 
544 
611 
677 
743 
809 
875 
941 

418 
485 
55i- 
617 
684: 
750 
816 
882 
948 

954 

961 

968 

974 

994 

*ooo 

*oo7 

*oi4 

82  020 
086 
151 
217 
282 
347 
413 
478 
543 

027 
092 
158 
223 
289 
354 
419 
484 
549 

033 
099 
164 
230 
295 
360 
426 

49  1 
556 

040 
105 

171 
236 
302 
367 
432 
497 
562 

046 

112 

I78 
243 
308 

373 
439 
504 
569 

060 
125 
191 
256 
321 
387 
452 
517 
5&2 

066 
132 
197 
263 

328 
393 
458 
523 
588 

073 
138 
204 
269 
334 
400 
465 
530 
595 

079 
145 

210 
276 
341 
406 
471 
536 

601 
666 

607 

614 

620 

627 

633 

646 

653 

659 

672 
737 
802 
866 
930 
995 
83  059 
123 
187 

679 
743 

808 
872 
937 

*OOI 

065 
129 
193 

685 
750 
814 
879 
943 
*oo8 
072 
136 
200 

692 
756 
821 
885 
950 
*oi4 
078 
142 
206 

698 
763 
827 
892 
956 

*020 
085 
149 
213 

705 
769 
834 
898 
963 

*02? 

091 
155 
219 

711 
776 
840 
905 
969 
*o33 
097 
161 
225 

718 
782 
847 
911 
975 
*040 
104 
1  68 
232 

724 
789 
853 
918 
982 
*046 
no 
174 
238 

730 
795 
860 
924 
988- 

*052 

117 

181 
245 

251 

257 

264 

270 

276 
340 
404 
467 
531 

594 
658 
721 
784 
847 
910 

973 
036 
098 
161 
223 
286 
348 
410 
473 

283 

289 

296 

302 

jo8_ 
372 
436 
499 
563 
626 
689 
753 
816 
879 

315 
378 
442 
506 
569 
632 
696 

759 
822 

321 
385 
448 
5" 
575 
639 
702 
765 
828 

327 
391 
455 
5i8 
582 
645 
708 
771 
835 
897 

334 
398 
461 
525 
588 
651 
715 
778 
841 

347 
410 
474 
537 
601 
664 
727 
790 
853 
916 

353 
417 
480" 
544 
607 
670 
734 
797 
860 

923 

359 
423 
487 
550 
613 
677 
740 
803 
866 

366 
429 
493 
556 
620 
683 
746 
809 
872 

885 

891 

904 

929 

935 

942 

948 
84  on 
073 
136 
198 
261 
323 
386 
448 

954 
017 
080 
142 
205 
267 
330 
392 
454 
5i6 

960 
023 
086 
148 

211 
273 
336 
398 
460 

522 

967 
029 
092 
155 
217 
280 
342 
404 
466 
528 

979 
042 
105 
167 
230 
292 
354 
417 
479 

985 
048 
in 
173 
236 
298 
361 
423 
485 

992 
055 
117 
1  80 
242 
305 
367 
429 
_49L 
553 

998 
06  r 
123 
186 
248 
3" 
373 
435 
497 
559 

*oo4 
067 
130 
192 
255 
317 
379 
442 
504 

5io 

535 

541 

547 

566 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

14 


LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P 

p. 

roo 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

RRRRRRRRR 

572 
634 
696 
757 
819 
880 
942 
8-3  003 
,065 

578 
640 
702 
763 
825 
887 
948 
009 
071 

584 
646 
708 
770 
831 
893 

954 
016 
077 

590 
652 

776 
837 
899 
960 

022 
083 

597 
658 
720 
782 
844 
905 
967 
028 
089 

603 

665 
726 
788 
850 
911 
973 
034 
095 

609 
671 
733 
794 
856 
917 

979 
040 

101 

615 

677 

739 
800 
862 
924 
985 
046 
107 

621 
683 
745 
807 
868 
930 
991 
052 
114 

628 
689 

813 
874 
936 
997 
058 

120 

2 

3 

r 

-•4 

no 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

4 

2.8 

711 
712 
713 
714 
715 

717 
718 
719 

187 
248 
309 
370 
431 
491 
552 

Cl2 
673 

193 
254 
315 

376 

437 
497 
558 
618 
679 

199 
260 
321 
382 
443 
503 
564 
625 
685 

205 
266 
327 

388 

449 
509 
570 
631 
691 

211 

272 

333 
394 

455 

576 
637 
697 

217 
278 

339 
400 
461 
522 
582 
643 
703 

224 
285 
345 
406 
467 
528 
588 
649 
709 

230 
291 

352 
412 

473 
534 
594 
655 
715 

236 
297 
358 
418 
479 
540 
600 
661 
721 

242 
303 
364 
425 
485 
546 
606 
667 
727 

6 
7 
8 
9 

4.2 
4.9 
5.6 
6-3 

r2o 

733 

739 

745 

751 

757 

763 

769 

775 

78i 

788 

721 

722 
723 
724 

725 
726 
727 

728 

729 

794 
854 
914 

974 
86  034 
094 
153 
213 
273 

800 
860 
920 
980 
040 

IOO 

159 
219 

279 

806 
866 
926 
986 
046 
106 
165 
225 
285 

812 
872 
932 
992 
052 

112 
171 
231 
291 

818 
878 
938 
998 
058 
118 
177 
237 
297 

824 
884 
^944 

064 
124 
183 
243 
303 

830 
890 
950 

*OIO 

070 
130 
189 

249 
308 

836 
896 
956 
*oi6 
076 
136 
195 
255 

842 
902 
962 

*022 
O82 
141 
201 
26l 
32O 

848 
908 
968 

*028 

088 
147 
207 

267 

326 

2 

3 
4 

I 
I 

Q 

1.2 
2-4 

36 

4.2 
4.8 
5-4 

rao 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

731 

732 

733 
734 
735 
736 
737 
738 
739 

392 
45i 

570 
629 

688 
747 
806 
864 

398 
-457 

576 
635 
694 

753 
812 
870 

404 
463 

522 

641 
700 
759 

817 
876 

410 
469 
528 
587 
646 
705 
764 
823 
882 

415 
475 
534 
593 
652 
711 
770 
829 
888 

421 
481 
540 
599 
658 
717 
776 
835 
894 

427 
487 
546 
605 
664 
723 
782 
841 
900 

433 
493 
552 
611 
670 

729 
788 
847 
906 

439 
499 
558 
617 
676 
735 
794 
853 
911 

445 
504 
564 
623 
682 
741 
800 
859 
917 

3 

5 

740 

923 

929 

935 

Q4I 

947 

953 

958 

964 

970 

976 

4 

2- 

741 
742 
743 
744 
745 
746 
747 
748 
749 

982 
87  040 
099 
157 
216 
274 
332 
390 
448 

988 
046 

221 
280 
338 
396 

454 

994 
052 
in 
169 
227 
286 
344 
402 
460 

999 
058 
116 
175 
233 
291 

349 
408 
466 

*oo5 
064 

122 

181 

239 
297 

355 
413 

*OII 

070 
128 
1  86 
245 
303 
361 
419 
477 

*oi? 
075 
134 
192 
251 
309 
367 
425 
483 

*023 

08  1 
140 

198 
256 

315 

373 
489 

*02g 
087 
146 
204 
262 
320 
379 
437 
495 

093 

210 
268 
326 
384 
442 
500 

6 
7 
8 
9 

3- 
4.0 
4-5 

rso 

506 

512 

5i8 

523 

529 

535 

54i 

547 

552 

558 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

LOGARITHMS. 


15 


N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

rso 

87  506 

512 

518 

523 

529 

535 

54i 

547 

552 

558 

751 

752 

753 
754 
755 
756 
757 
758 
759 

564 
622 
679 

737 
795 
852 
910 
967 
88  024 

570 
628 
685 

743 
800 
858 
915 
973 
030 

576 
633 
69I 

749 
806 
864 
921 
978 
036 

581 
639 

697 

754 
812 
869 
927 
984 
041 

587 
645 
703 
760 
818 
875 
933 
990 
047 

593 
651 
708 
766 
823 
881 
938 
996 
053 

599 
656 
714 
772 
829 
887 
944 

*OOI 

058 

604 
662 
720 

777 
835 
892 
950 

*007 

064 

610 

668 
726 
783 
841 
898 
955 
*oi3 
070 

616 
674 
731 
789 
846 
904 
961 
*oi8 
076 

r60 

08  1 

087 

093 

098 

104 

no 

116 

121 

127 

133 

761 
762 
763 
764 
765 
766 
767 
768 
769 

138 
195 
252 
3°9 
366 
423 
480 
536 
593 

144 

201 

258 
315 
372 
429 
485 
542 
598 

150 
207 
264 
321 
377 
434 
491 
547 
604 

156 
213 
270 
326 
383 
440 
497 
553 
610 

161 

218 
275 
332 
389 
446 
502 
559 
615 

167 
224 
281 
338 
395 
45i 
508 
564 
621 

173 
230 

287 
343 
400 
457 
513 
570 
627 

178 
235 
292 

349 

406 

463 
519 

5/6 
632 

184 
241 
298 
355 
412 
468 
525 
58i 
638 

190 

247 
304 
360 
417 
474 
530 
587 
643 

6 

3   '^8 
4   2.4 
5   3-° 
6   3.6 

I  ::S 

9   5-4 

rro 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

771 

772 

773 
774 
775 
776 
777 
778 
779 

705 
762 
8x8 

874 
930 
986 
89  042 
098 
i54 

711 

767 
824 
880 
936 
992 
048 
104 
159 

717 

773 
829 
885 
941 
997 
053 
109 
165 

722 
779 
835 
891 
947 
*oo3 

059 
H5 
170 

728 
784 
840 
897 
953 

*009 

064 

120 
I76 

734 
790 
846 
902 
958 
*oi4 
070 
126 
182 

739 
795 
852 
908 
964 

*020 
076 
.131 

I87 

745 
801 
857 
913 
969 

*025 

08  1 
137 
193 

750 
807 
863 
919 
975 
031 
087 
143 
198 

756 
812 
868 
925 
981 
*o37 
092 
148 
204 

rso 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

78i 
782 
783 
784 
785 
786 
787 
788 
789 

265 
321 
376 
432 
4P7 
542 
597 
653 
708 

271 
326 

382 

437 
492 
548 
603 
658 
•713 

276 
332 

387 

443 
498 
553 
609 
664 
719 

282 
337 
393 
448 
504 
559 
614 
669 
724 

287 

343 

398 
454 
509 
564 
620 
675 
73° 

293 

348 
404 
459 
515 
570 
625 
680 
735 

298 

354 
409 
465 
520 
575 
631 
686 
741 

304 

360 

415 
470 

526 

581 

636 

691 

746 

310 
365 
421 
476 
531 
586 
642 
697 
752 

315 
37i 
426 
481 
537 
592 
647 
702 
757 

5 

3   *• 

S   2. 
6   3- 

1  \-l 

9  4-5 

r90 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

791 
792 
793 
794 
795 
796 
797 
798 
799 

818 
873 
927 
982 
90  037 
091 
146 

200 
255 

823 
878 
933 
988 
042 
097 
151 
206 
260 

829 
883 
938 
993 
048 

102 
157 
211 
266 

834 
889 
944 
998 
053 
108 
162 
217 
271 

840 
894 
949 
*oo4 

059 
H3 
168 

222 
276 

845 
900 

*955 
*OO9 

064 
119 
173 

227 
282 

851 
905 

96° 
*oi5 
069 
124 
179 
233 
287 

856 
911 

*966 

*020 
075 
I29 
I84 
238 
293 

862 
916 
971 

*026 

080 
135 

189 

244 

298 

867 
922 
977 
*03i 
086 
140 
195 
249 
304 

800 

-309 

3H 

320 

325 

33t 

336 

342 

347 

352 

358 

N. 

L.   0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

16 


LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

800 

9o  309 

3U 

320 

325 

33i 

336 

342 

347 

352 

358 

80  1 
802 
803 
804 
805 
806 
807 
808 
809 

363 
417 
472 
526 
580 
634 
687 
741 
795 

369 
423 
477 
53i 
585 
639 
693 
747 
800 

374 

428 
482 
536 
59° 
644 
698 
752 
806 

380 
434 
488 
542 
596 
650 
703 
757 
811 

385 
439 
493 
547 
601 
655 
709 
763 
816 

390 
445 
499 
553 
607 
660 
7H 
768 
822 

396 
45o 
504 
558 
612 
666 
720 
773 
827 

401 

455 
509 
563 
617 
671 
725 
779 
832 

407 
461 

515 
569 
623 
677 
730 
784 
838 

412 

466 
520 
574 
628 
682 
736 
789 
843 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

811 
812 
813 
814 
8i5 
816 
817 
818 
819 

902 
956 
91  009 
062 
116 
169 

222 

275 
328 

907 
961 
014 
068 

121 
174 
228 
28l 

334 

913 
966 
020 
073 
126 
1  80 
233 
286 

339 

918 
972 
025 
078 
132 
185 
238 
291 
344 

924 
977 
030 
084 
137 
190 
243 
297 
350 

929 
982 
036 
089 
142 
196 
249 
302 
355 

934 
988 
041 
094 
148 

2OI 
254 

307 
360 

940 
993 
046 

100 

153 
206 

259 
312 
365 

945 
998 
052 
105 

158 

212 

265 
318 
371 

950 
*oo4 
057 
no 

164 

217 

270 
323 

376 

6 

2    1.2 

3   1.8 
4   2.4 

1  l°6 

7   4-2 
8  4.8 
9   5-4 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

821 
822 
823 
824 
825 
826 
827 
828 
829 

434 
487 
540 
593 
645 
698 
751 
803 
855 

440 
492 
545 
598 
65i 
703 
756 
808 
861 

445 
498 
551 
603 
656 
709 
761 
814 
866 

450 
503 
556 
609 
661 
7H 
766 
819 
871 

455 
508 
56i 
614 
.666 
719 
772 
824 
876 

461 
514 
566 
619 
672 
724 
777 
829 
882 

466 
519 
572 
624 
677 
730 
782 
834 
887 

471 
524 

577 
630 
682 
735 
787 
840 
892 

477 
529 
582 
635 
687 
740 
793 
845 
897 

482 
535 
587 
640 
693 
745 
798 
850 
003 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

831 
832 
833 
834 
835 
836 
837 
838 

839 

960 

92  012 
065 
II? 
I69 
221 
273 
324 
376 

965 
018 
070 

122 

!?4 
226 
278 
330 

381 

971 
023 
075 
127 

179 
231 

283 
335 
387 

976 
028 
080 

IS 

236 

288 
340 
392 

981 
033 
085 
137 
189 
241 
293 
345 
397 

986 
038 
091 
143 
195 
247 
298 
350 
402 

991 
044 
096 
148 
200 
252 
304 
355 
407 

997 
049 

IOI 

153 
205 
257 
309 
361 
412 

*002 
054 
1  06 
158 
210 
262 
3M 

306 
418 

*oo7 

059 
in 
163 
215 
267 

319 
371 
423 

* 

2. 

:   2. 

I: 

g   4- 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

841 
842 
843 
844 
845 
846 
847 
848 
849 

480 
531 
583 
634 

686 
737 
788 
840 
891 

485 
536 
588 

639 
691 
742 
793 

845 
896 

49° 
542 
593 
645 
696 
747 
799 
850 
901 

495 
547 
598 
650 
701 
752 
804 
855 
906 

500 

552 
603 
655 
706 
758 
809 
860 
911 

505 
557 
609 
660 
711 
763 
814 
865 
916 

5" 
562 
614 
665 
716 
768 
819 
870 
921 

5i6 
567 
619 
670 
722 
773 
824 
875 
927 

52i 
572 
624 
675 
727 
778 
829 
881 
932 

526 

578 
629 
681 
732 
783 
834 
886 
937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

<)** 

N. 

L.  o 

; 

2 

3 

4 

5 

6 

7 

8  1 

9 

P.  P. 

LOGARITHMS. 


17 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

851 
852 
853 
854 
855 
856 
857 
858 

859 

993 
93  044 
095 
146 
197 
247 
298 
349 
399 

998 
049 

100 

151 

202 
252 
303 

354 
404 

*003 

054 
105 
156 
207 

258 
308 

359 
409 

*oo8 

059 
no 
161 

212 

263 

313 

364 

414 

*oi3 
064 

1  66 
217 
268 

369 
420 

*oi8 
069 

120 

171 

222 
273 
323 

374 

425 

*024 

075 
125 

176 

227 

278 
328 

379 
43° 

080 
131 

181 
232 
283 
334 
384 
435 

*034 
085 
136 
186 
237 
288 
339 
389 
440 

*039 
090 
141 
192 
242 
293 
344 
394 
445 

2 

3 

6 

1.2 

860 

450 

455 

460 

465 

470 

475 

480 

485 

49° 

495 

4 
5 

2.4 

861 
862 
863 
864 
865 
866 
867 
868 
869 

500 
551 
60  1 
651 
702 
752 
802 
852 
902 

505 
556 
606 
656 
707 
757 
807 
857 
907 

56i 
611 
66  1 
712 
762 
812 
862 

515 

566 

616 
666 
717 
767 
8i7 
867 
917 

520 

621 
671 

722 
772 
822 
872 
922 

526 
576 
626 
676 
727 
777 
827 
877 
927 

631 
682 
732 

782 
832 
882 
932 

536 
586 
636 
687 
737 
787 
837 
887 
937 

541 
591 
641 
692 
742 
792 
842 
892 
942 

546 
596 
646 
697 
747 
797 
847 
897 
947 

6 
7 
8 
9 

4-2 

5-4 

sro 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

8?2 

873 
874 

875 

876 

877 

878 
879 

94  002 
052 

IOI 

201 
250 
300 

349 
399 

007 
057 
106 
156 
206 
255 
305 
354 
404 

012 
062 
III 

161 

211 
260 
310 

359 
409 

017 
067 
116 
1  66 
216 
265 
315 
364 
414 

022 
072 
121 

221 
270 
320 
369 
419 

027 
077 
126 
176 
226 
275 
325 
374 
424 

032 
082 

181 
231 
280 
330 
379 
429 

037 
086 
136 
1  86 
236 
285 
335 
384 
433 

042 
091 
141 
191 
240 
290 
340 
389 
438 

047 
096 
146 
196 
245 
295 
345 
394 
443 

a 
3 
4 

i 
I 

9 

5. 

10-  '    ) 

'5  y 

2  0. 
*J> 

3  5 
4  o 
4  5 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

881 
882 
883 
884 
885 
886 
887 
888 
889 

49s 
547 
596 
645 
694 
743 
792 
841 
890 

503 
552 
601 
650 
6c; 
748 
797 
846 
895 

507 
557 
606 
655 
704 
753 
802 
851 
900 

512 
562 
611 
660 
709 
758 
807 
856 
905 

517 
567 

616 
665 

763 
8'* 
861 
910 

522 

621 
670 
719 

768 
,817 
$66 
915 

527 
5/6 
626 
675 
724 
773 
822 
871 
919 

532 

630 
680 
729 

778 
827 
876 
924 

537 
586 
635 
685 
734 
783 
832 
880 
929 

542 

640 

689 
738 
787 
836 
885 
934 

2 

3 

4 

1.2 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

4 

1.6 

891 
892 
893 
894 
895 
896 
897 
898 
899 

988 
95  036 
085 
134 
182 
231 
279 
328 
376 

993 
041 
090 
139 
187 
236 
284 
332 

998 
046 
095 
143 
192 
240 
289 
337 
386 

*002 

051 

100 

148 
197 
245 
294 

342 
390 

056 
105 
153 

202 

250 
299 

347 
395 

*OI2 
06  1 
109 
158 
207 
255 
303 
352 
400 

*oi7 
066 
114 
163 

211 
260 
308 

357 
405 

*022 

071 
119 
168 
216 
265 
313 
361 
410 

*027 

075 

124 
173 

221 
270 
318 
366 
415 

*032 
O80 
129 
177 
226 
274 
323 
371 
419 

6 
7 
8 
9 

1:3 

1:2 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

p. 

18 


LOGARITHMS. 


N. 

L.  o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

900 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

901 
902 
903 
904 
9°5 
906 
907 
908 
909 

472 
52i 
569 
617 
665 
713 
761 
809 
856 

477 
525 
574 
622 
670 
718 
766 
813 
861 

482 
530 
578 
626 
674 
722 
770 
818 
866 

487 
535 
583 
631 
679 
727 
775 
823 
871 

492 
540 
588 
636 
684 
732 
780 
828 
875 

497 

545 
593 
641 
689 

737 
785 
832 
880 

501 
550 
598 
646 
694 
742 
789 
837 
885 

506 
554 
602 
650 
698 
746 
794 
842 
890 

5" 

559 
607 

655 
703 
75i 
799 

847 
895 

5i6 
564 
612 
660 
708 
756 
804 
852 
899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

5 

911 
912 

913 
914 
915 
916 
917 
918 
919 

952 
999 
96  047 

095 
142 
190 
237 
284 
332 

957 
*oo4 
052 
099 
147 
194 
242 
289 
336 

961 
*oog 
057 
104 
152 
199 
246 
294 
34i 

966 
*oi4 
061 
109 
156 
204 
251 
298 
346 

971 
*org 
066 
114 
161 
209 
256 
303 
350 

976 

*023 

071 
118 
1  66 
213 
261 
308 
355 

980 

*028 

076 
123 
i?i 
218 
265 
313 
360 

985 
*o33 
080 
128 
175 
223 
270 
317 
365 

990 
*038 
085 
133 
1  80 
227 
275 
322 
369 

995 

*042 

090 
137 
185 
232 
280 
327 
374 

a   i. 
3   *• 
4   2- 

5   2- 

6   3- 

8   4' 
9  4- 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

921 
922 
923 
924 
925 
926 
927 
928 
929 

426 
473 
520 
567 
614 
661 
708 
755 
802 

43i 
478 
525 
572 
619 
666 
713 
759 
806 

435 
483 
530 
577 
624 
670 
717 
764 
8n 

440 
487 
534 
58i 
628 
675 
722 
769 
816 

445 
492 

539 
586 

633 
680 
727 
774 
820 

450 
497 
544 
591 
638 
685 
731 
778 
825 

454 
5oi 
548 
595 
642 
689 
736 
783 
830 

459 
506 
553 
600 
647 
694 
74i 
788 
834 

464 
5" 
558 
605 
652 
699 
745 
792 
839 

468 
515 
562 
609 
656 
703 
750 
797 
844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

93i 
932 
933 
934 
935 
936 
937 
938 
939 

895 
942 
988 
97  035 
08  1 
128 
174 

220 
267 

900 
946 
993 
039 
086 
132 
179 
225 
271 

904 
95i 
997 
044 
090 
137 
183 
230 
276 

909 

956 

*002 
049 
095 
142 

188 
234 
280 

914 

960 
*oo7 
053 

100 

146 
192 
239 

285 

918 
965 

*OII 

058 
104 
151 
197 
243 
290 

923 
970 
*oi6 
063 
109 
155 

202 
248 
294 

928 
974 

*02I 
067 
114 

160 

206 
253 
299 

932 

979 

*025 

072 
118 
165 

211 

257 
304 

937 
984 
*030 
077 
123 
169 
216 
262 
308 

4 
i  0.4 

2    0.8 

I  ^ 
1  - 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

8   3-J 

9   3-6 

941 
942 
943 
944 
945 
946 
947 
948 

949 

359 
405 
45i 
497 
543 
589 

727 

364 
410 
456 
502 
548 
594 
640 
685 
731 

368 
414 
460 
506 
552 
598 
644 
690 
736 

373 
419 
465 
5ii 
557 
603 
649 
695 
740 

377 
424 
470 
5i6 
562 
607 
653 
699 
745 

382 
428 
474 
520 
566 
612 
658 
704 
749 

387 

433 
479 
525 
571 
617 
663 
708 
754 

39i 
437 
483 
529 
575 
621 
667 
713 
759 

396 
442 
488 

534 
580 
626 
672 
717 
763 

400 
447 
493 
539 
585 
630 
676 
722 
768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

8i3 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGARITHMS. 


19 


N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

95i 
952 
953 
954 
955 
956 
957 
953 
959 

818 
864 
909 
955 
98  ooo 
046 

091 
137 
182 

823 
868 
914 

959 
005 
050 
096 
141 
186 

827 
873 
9I8 
964 
009 
055 
100 

146 
191 

832 
877 
923 
968 
014 
059 
105 
150 
i95 

836 
882 
928 
973 
019 
064 
log 
155 

200 

84I 
886 
932 
978 
023 
068 
114 

159 
204 

845 
891 
937 
982 
028 
073 
118 
164 
209 

850 
896 
941 
987 
032 
078 
123 
168 
214 

855 
900 
946 
991 
037 
082 
127 
i?3 
218 

859 
905 
95o 
996 
041 
087 
132 
i?7 
223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

961 
962 
963 
964 
965 
966 
967 
968 
969 

272 

318 
363 

408 
453 
498 
543 
588 
632 

277 
322 
367 
412 
457 
502 
547 
592 
637 

281 
327 
372 
417 

462 

507 

552 

597 
641 

286 
33i 
376 
421 
466 
5" 
556 
60  1 
646 

290 
336 
381 
426 
471 
5I6 
561 
605 
650 

295 
340 
385 
430 
475 
520 
565 
610 
655 

299 
345 
390 
435 
480 
525 
570 
614 
659 

304 
349 
394 
439 
484 
529 
574 
619 
664 

308 
354 
399 
444 
489 
534 
579 
623 
668 

313 
358 
403 
448 
493 
538 
583 
628 
673 

5 

2    I. 

3  i- 

i  i: 

I  I: 

9   4- 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

97i 
972 
973 
974 
975 
976 
977 
978 

979 

722 
767 
8n 
856 
900 
945 
989 
99  034 
078 

726 
771 
816 
860 
905 
949 
994 
038 
083 

731 
776 
820 
865 
909 
954 
998 
043 
087 

735 
780 
825 
869 
914 
958 
*oo3 
047 
092 

740 
784 
829 
874 
9I8 
963 

*oo7 
052 
096 

744 
789 
834 
878 
923 
967 

*OI2 
056 
IOO 

749 
793 
838 
883 
927 
972 
*oi6 
06  1 
105 

753 
798 
843 
887 
932 
976 

*02I 
065 
I09 

758 
802 
847 
892 
936 
981 

*025 

069 

114 

762 
807 
851 
896 
941 
985 

*O2g 

074 

118 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

981 
982 

983 
984 
985 
986 
987 
988 
989 

167 

211 

255 
300 

344 
388 
432 
476 
520 

i?i 
216 
260 
304 

348 
392 
436 
480 

524 

176 
220 

264 
308 
352 
396 

441 

484 
528 

1  80 
224 
269 

313 
357 
401 
445 
489 
533 

185 
229 
273 
317 
361 
4°5 
449 
493 
537 

l8g 
233 
277 
322 
366 
410 

454 
498 
542 

193 
238 
282 
326 
370 
414 
458 
502 
546 

198 
242 
286 
330 

374 
419 
463 
506 
550 

202 
247 
29I 

335 
379 
423 
467 
Sir 
555 

207 
251 
295 
339 
383 
427 
471 
515 
559 

4 

2   ois 

\    \'l 
5   •!•<> 

6   2.4 
7   2.8 
8   3.2 
9  3-6 

990 

564 

568 

572 

577 

58i 

585 

590 

594 

599 

603 

991 
992 
993 
994 
995 
996 
997 
998 

999 

607 
651 
695 
739 
782 
826 
870 
913 
957 

612 
656 
699 
743 
787 
830 
874 
917 
961 

616 
660 
704 
747 
791 
835 
878 
922 
965 

621 
664 
708 
752 
795 
839 
883 
926 
970 

625 
669 
712 
756 
800 
843 
887 
930 
974 

629 
673 
717 
760 
804 
848 
891 
935 
978 

634 
677 
721 
765 
808 
852 
896 
939 
983 

638 
682 
726 
769 
8i3 
856 
900 
944 
987 

642 
686 
730 
774 
817 
861 
904 
948 
991 

647 
691 
734 
778 
822 
865 
909 
952 
996 

1000 

00  000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

N. 

L.  o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

TABLES 


OF 

NATURAL  SINES,   COSINES, 

TANGENTS, 
AND   COTANGENTS 

GIVING  THE  VALUES  OF  THE  FUNCTIONS  FOR 
ALL  DEGREES  AND  MINUTES  FROM 

O°  TO  QO° 


NATURAL  SINES  AND  COSINES. 


0 

I 

0 

2 

0 

3 

4 

0 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

.00000 

.00029 

•01745 
.01774 

.99985 
.99984 

.03490 
•03519 

•99939 
.99938 

•05234 
.05263 

.99863 
.99861 

.06976 
.07005 

•99756 
•99754 

Go 

TO 

.00058 

.01803 

.99984 

.03548 

•99937 

.05292 

.99^.60 

.07034 

•99752 

5« 

.00087 

.01832 

•99983 

•°3577 

.99936 

.05321 

.99858 

•07063 

.99750 

=,7 

.00116 

.01862 

.99983 

.03606 

•99935 

•05350 

•99857 

.07092 

•99748 

56 

.00145 

.01891 

.99982 

•03635 

•99934 

•05379 

•99855 

.07121 

.99746 

55 

.00175 

.01920 

.99982 

.03664 

•99933 

.05408 

.99854 

.07150 

-99744 

54 

.00204 

.01949 

.99981 

•03693 

•99932 

•05437 

•99852 

.07179 

.99742 

53 

.00233 

.01978 

.99980 

.03723 

•99931 

.05466 

.99851 

.07208 

•99740 

52 

.00262 

.02007 

.99980 

.03752 

.99930 

•05495 

.99849 

.07237 

•99738 

51 

1 

.00291 

•99979 

•03781 

•99929 

.05524 

•99847 

.07266 

.99736 

50 

, 

.00320 

.99999 

.02065 

•99979 

.03810 

•99927 

•05553 

.99846 

.07295 

•99734 

4" 

I 

.00349 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

•99731 

4S 

I 

.00378 

.99999 

.02123 

•99977 

.03868 

•99925 

.05611 

.99842 

•°7353 

-99729 

47 

I. 

.00407 

.99999 

.02152 

•99977 

.03897 

•99924 

.05640 

.99841 

.07382 

.99727 

4" 

I  : 

.00436 

.02181 

.99976 

.03926 

.99923 

.05669 

•99839 

.07411 

•99725 

45 

I 

.00465 

.99999 

.O22II 

.99976 

•03955 

.99922 

.05698 

•99838 

.07440 

.99723 

44 

1? 

.00495 

.99999 

.02240 

•99975 

.03984 

.99921 

•05727 

.99836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

•99974 

.04013 

.99919 

•09834 

.07498 

.99719 

42 

'9 

•00553 

.99998 

.02298 

•99974 

.04042 

.99918 

.05785 

•99833 

•07527 

.99716 

41 

.00582 

.99998 

.02327 

•99973 

.04071 

.99917 

.05814 

.99831 

•07556 

.09714 

4'> 

21 

.00611 

.99908 

.02356 

.99972 

.04100 

.99916 

•05844 

.99829 

•07585 

.99712 

39 

22 

.00640 

.99998 

•02385 

.99972 

.04129 

•99915 

•05873 

.99827 

.07614 

.99710 

j8 

23 

.00665 

.99998 

.02414 

.99971 

.04159 

.99913 

.05902 

.99826 

.07643 

.99708 

37 

24 

.00698 

.99998 

.02443 

.99970 

.04188 

.99912 

•Q5931 

.99824 

.07672 

•99705 

3<- 

25 

.00727 

.09997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

•99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.09701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819 

•07759 

.99699 

33 

29 

.00814 
.00844 

.99997 
.99996 

.02560 
.02589 

.90967 
.99966 

.04304 
•04333 

.99907 
.99906 

.06047 
.06076 

.99817 
.99815 

.07788 
.07817 

.99696 
.99694 

32 
31 

3° 

.00873 

.99996 

.026X8 

.99966 

.04362 

•99905 

.06105 

.99813 

.07846 

.99692 

3° 

31 

32 

.00902 
.00931 

.99996 
.99996 

.02647 
.02676 

.99965 
•99964 

.04391 
.04420 

.99904 
.99902 

.06134 
.06163 

.99812 
.99810 

•07875 
.07904 

.99689 
.99687 

3 

33 

.00960 

•99995 

.02705 

•99963 

.04449 

.99901 

.06192 

.99808 

•07933 

•99685 

27 

34 

.00989 

•99995 

•02734 

•99963 

.04478 

.09000 

.06221 

.99806 

.07962 

.99683 

20 

P 

.01047 

•99995 
•99995 

.02763 
.02792 

.99962 
.99961 

.04507 
•04536 

.99898 
•99897 

.06250 
.06279 

.99804 
.99803 

.07991 
.08020 

.99680 
.99678 

25 
24 

57 

.01076 

.99994 

.O282I 

.99960 

•04565 

.99896 

.06308 

.99801 

.08049 

.99676 

23  • 

38 

.01105 

•99994 

.02850 

•99959 

•04594 

•99894 

•06337 

•99799 

.08078 

.99673 

39 

.01134 

•99994 

.02879 

.99959 

.04623 

•99893 

.06366 

•99797 

.08.07 

.99671 

21 

.01164 

•99993 

•99958 

•04653 

.99892 

.06395 

•99795 

.08,36 

.99668 

20 

4i 

.01193 

•99993 

.02938 

•99957 

.04682 

.99890 

.06424 

•99793 

.08165 

.99666 

9 

42 

.01222 

•99993 

.02967 

•99956 

.04711 

.99889 

.06453 

•99792 

.-08194 

.99664 

a 

43 

.OI25I 

.99992 

.02996 

•99955 

.04740 

.99888 

.06482 

.99790 

.08223 

.99661 

7 

44 

.01280 

•99992 

.03025 

•99954 

•04769 

.99886 

.06511 

.99788 

.08252 

.99650 

(i 

45 

•01309 

.99991 

•03054 

•99953 

.04798 

•99885 

.06540 

.99786 

.08281 

•99657 

5 

4f> 

.0  338 

•99991 

•03083 

•99952 

.04827 

.99883 

.06569 

.99784 

.08310 

•99654 

4 

47 

.0  367 

.99991 

.99952 

.04856 

.99882 

.06598 

.99782 

•08339 

.99652 

i 

48 

.0  396 

•99990 

.03141 

•99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

« 

.0  425 

•99990 

.03170 

.99950 

.04914 

.99879 

.06656 

•99778 

.08397 

•99647 

i 

50 

.0  454 

.99989 

.03109 

•99949 

.04943 

.99878 

.06685 

.99776 

.08426 

•99644 

0 

Si 

•  0483 

.09989 

.03228 

.99948 

.04972 

•99876 

.06714 

•99774 

•08455 

.99642 

g 

52 

•o  5'3 

.99989 

•03257 

•99947 

.05001 

•99875 

.06743 

.99772 

.08484 

.99639 

§ 

53 
54 

.0  57! 

.99988 
.99988 

.03286 
.03316 

.99946 
•99945 

.05030 
•05059 

•99873 
.99872 

.06773 
.06802 

.99770 
.99768 

.08513 
.08542 

•99637 
•99D35 

7 
6 

55 

.0  600 

.99987 

•03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

5* 

.0  629 

•99987 

•03374 

•99943 

.05117 

.99869 

.06860 

.99764 

.08600 

.99630 

4 

=57 

.0  658 

.99986 

.99942 

.05146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

.0687 

.99986 

•03432 

.99941 

•05175 

.99866 

.06918 

.99760 

.08658 

.99625 

2 

59 
60 

.0  716 

.99985 
.99985 

.03461 
.03490 

.99940 
•99939 

.05205 
.05234 

.90864 
.99863 

.06947 
.06976 

•99758 
•99756 

.08687 
.08716 

.09622 
.99619 

,', 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

8 

f 

8 

\° 

8 

f° 

8 

8 

,0 

NATURAL  SINES  AND  COSINES. 


5 

0 

6 

0 

7 

0 

8 

0 

9 

o 

Sine 

Cosine 

S  ne 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

11 

.08716 
.08745 

.99619 
.99617 

•10453 
.  10482 

•99452 

•99449 

.12187 
.12216 

•99255 
•99251 

.13917 
.13946 

.99027 
.99023 

•15643 
.15672 

.98769 
.98764 

60 

2 

.08774 

.99614 

.10511 

•99446 

.12245 

.99248 

•13975 

.99019 

.15701 

.98760 

S<"> 

3 

.08803 

.99612 

•  10540 

•99443 

•  12274 

•  99244 

.14004 

.99015 

•1573° 

•98755 

^7 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.  99240 

•  14033 

.99011 

•15758 

•98751 

56 

.08860 

.99607 

•  10597 

•99437 

.12331 

•99237 

.1406! 

.99006 

•15787 

.98746 

ss 

6 

.08889 

.99604 

.10626 

•99434 

.12360 

•99233 

.14000 

.99002 

.15816 

•98741 

54 

7 

.08918 

.99602 

.10655 

•99431 

•12389 

.09230 

.98998 

•  15845 

•98737 

5  \ 

8 

.08947 

•99599 

.  10684 

.99428 

.12418 

.99226 

.14148 

•98994 

15873 

•98732 

=,-'- 

g 

.08976 

.99596 

.10713 

.99424 

•  12447 

.99222 

.14177 

98990 

.15902 

.98728 

=,' 

10 

.09005 

•99594 

.  10742 

.99421 

.12476 

.99219 

.  14205 

.98986 

•15931 

.98723 

,, 

•09034 

•99591 

.10771 

.99418 

.  12504 

•99215 

•14234 

.98982 

•"5959 

.98718 

49 

12 

.09063 

.99588 

.10800 

.99415 

•12533 

.99211 

.14263 

.98978 

.  15988 

•98714 

48 

13 

.09092 

.99586 

.  10829 

.99412 

.I2562 

.99208 

.14292 

.98973 

.  16017 

.98709 

47 

.09121 

•99583 

.  10858 

.99409 

•12591 

.99204 

•  14320 

.98969 

.16046 

.98704 

4" 

IS 

•09150 

.99580 

.  10887 

.99406 

.  12620 

.99200 

•14349 

.98965 

.16074 

.98700 

41 

If, 

.09179 

•99578 

.  10916 

.99402 

.  12649 

.99197 

•14378 

.9896! 

.16103 

•98695 

44 

17 

.09208 

•99575 

.10945 

•99399 

.12678 

•99193 

.14407 

•  98957 

.16132 

.98690 

4  i 

t8 

.09237 

•99572 

•10973 

.99396 

.12706 

.99189 

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.98953 

.16160 

.98686 

19 

.09266 

•99570 

.  IOO2 

•99393 

•12735 

.99186 

.14464 

.98948 

.16189 

.98681 

4' 

20 

.09295 

•99567 

.  1031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

22 

•09324 
•09353 

.99564 
.99562 

.  1O6O 
.   089 

.99386 
•99383 

•  "793 

.12822 

.99178 

•99*75 

•14522 
•  M55i 

.98940 
.98936 

.16246 
.16275 

.98671 
.98667 

a 

23 

.09382 

•99559 

.  118 

.90380 

.12851 

.99171 

.14580 

.98931 

•16304 

.98662 

37 

24 

.09411 

.90556 

•  M7 

•99377 

.  12880 

.99167 

.14608 

.98927 

•  J6333 

.98657 

36 

.09440 

•99553 

.  176 

•99374 

.12908 

.99163 

•14637 

.98923 

.16361 

.98652 

35 

2f> 

.09469 

•99551 

•  205 

•99370 

•12937 

.99160 

.14666 

.98919 

.16390 

.98648 

34 

27 

.09498 
•09527 

•99548 
•99545 

:  203 

•09367 
•99364 

.!2966 

•  12995 

.99156 
•99152 

.14695 
.14723 

.98914 
.98910 

.16419 
•  16447 

.98643 
.98638 

33 

32 

ag 

•09556 

•99542 

.  291 

.99360 

.13024 

.99148 

•14752 

.98906 

.!6476 

•98633 

n 

3" 

•09585 

.99540 

.  320 

•99357 

•13053 

.99144 

.14781 

.98902 

.  16505 

.98629 

3° 

31 

.09614 

•99537 

•  349 

•99354 

.13081 

.99141 

.14810 

.98897 

•16533 

.98624 

20 

32 

.09642 

•99534 

•  378 

•99351 

.13110 

•99137 

.14838 

.98893 

.  16562 

.98619 

2.S 

33 

.09671 

•99531 

•  407 

•99347 

•13139 

•99133 

.14867 

•98889 

.  16591 

.98614 

'-'-7 

34 

.09700 

.99528 

436 

•99344 

.13168 

.99129 

.14896 

.  16620 

.98609 

26 

35 

.09729 

.99526 

465 

•99341 

•13197 

.99125 

•14925 

!g888o 

.16648 

.98604 

as 

36 

.09758 

•99523 

•   494 

•99337 

.13226 

.99122 

•  M954 

.98876 

.  16677 

.98600 

37 

.09787 

.99520 

•   523 

•99334 

•13254 

.99118 

.  14982 

.98871 

.16706 

•98595 

23 

39 

.09816 
.09845 

•99517 
.99514 

•99331 
.99327 

.13283 

.13312 

.99114 
.99110 

.15011 
.  15040 

.98867 
.98863 

'.%£ 

•98590 
•98585 

22 
21 

40 

.09874 

•995" 

.'   609* 

•99324 

•13341 

.99106 

•15069 

.98858 

.  16792 

.98580 

30 

41 

.09903 

.99508 

•   638 

.99320 

•13370 

.99102 

•15097 

.98854 

.16820 

•98575 

I" 

42 

.09932 

.99506 

.   667 

•99317 

•13399 

.99098 

•15126 

.98849 

.  16849 

.98570 

i 

43 

.09961 

•99503 

.   696 

•993M 

-  13427 

.99094 

•iS'55 

.98845 

.16878 

•98565 

7 

44 

.09990 

.99500 

•   725 

.99310 

•13456 

.99091 

.15184 

.98841 

.16906 

•98561 

(, 

45 

•99497 

•  754 

•99307 

•13485 

.99087 

.15212 

.98836 

.  16935 

•98556 

9 

*6 

.10048 

•99494 

•  783 

•09303 

•13514 

•99083 

.15241 

•98832 

.16964 

•98551 

4 

47 

.10077 

.99491 

.  812 

.99300 

•  13543 

.99079 

.15270 

.16992 

•98546 

3 

,,S 

.10106 

.99488 

.  840 

•99297 

•13572 

•99°75 

•  15299 

.98823 

.17021 

.98541 

a 

49 

.10135 

.99485 

.  869 

.99293 

.13600 

.99071 

•i5327 

.98818 

.17050 

.98536 

i 

5" 

.  10164 

.99482 

.  898 

.99290 

.  13629 

•99067 

•15356 

.98814 

.17078 

•08531 

" 

5I 

.10192 

•99479 

•  927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

•98526 

• 

52 

.  I022I 

•99476 

•  956 

•99283 

•13687 

•99059 

•  i54M 

.98805 

.1  .36 

.98521 

B 

53 

.  10250 

•99473 

•  985 

.99279 

.137,6 

•99°55 

•15442 

.98800 

.1  164 

.98516 

7 

54 

.  10279 

•99470 

.99276 

•13744 

.99051 

•15471 

.98796 

•i  193 

.98511 

6 

55 

.10308 

•99467 

•  043 

.99272 

•13773 

.99047 

.15500 

.98791 

.1  222 

.98506 

5 

56 

•  10337 

•99464 

.  071 

.99269 

.!3802 

.99043 

.15529 

.98787 

.1  250 

.98501 

4 

57 

.10366 

.99461 

IOO 

.99265 

.  13831 

.99039 

•15557 

.98782 

•I  279 

.98496 

.-i 

58 

•  I03Q5 

•99458 

129 

.99262 

.13860 

.99035 

•15586 

•98778 

•I  3°8 

.98491 

2 

59 

.  10424 

•99455 

158 

.99258 

.  13889 

•99031 

•15615 

•98773 

•17336 

.98486 

I 

60 

•  10453 

•99452 

187 

.99255 

•13917 

.90027 

.15643 

•98769 

.17365 

.98481 

Q 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

8, 

f 

8. 

5° 

8 

2° 

8 

:° 

8< 

5° 

NATURAL  SINES  AND  COSINES. 


I 

o° 

I 

i  ° 

I 

2° 

1 

3° 

i 

4° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

•17365 
•17393 

.98481 
.98476 

.  908. 

.  9109 

.98  63 

•98  57 

.20791 
.20820 

•97815 
.97809 

.22495 
•22523 

•97  37 
•97  3° 

.24192 
.  24220 

.97030. 
•97023 

Go 

5V 

3 

.17422 
•'7451 

•98471 
.98466 

.  9138 
•  9167 

.98  S2 

.98  46 

.20848 
.20877 

.97803 

•97797 

•22552 
.22580 

•97  24 
•97  i? 

.24249 
•24277 

•97015 
.97008 

58 
57 

4 

•17479 

.98461 

•  9'95 

.98  40 

.20905 

•97791 

.22608 

•97  'I 

•  24305 

.97001 

5 

.17508 

•98455 

•  9224 

•98  35 

•20933 

.97784 

.22637 

.97404 

•24333 

.96994 

55 

6 

•'7537 

.08450 

•  9252 

.98  29 

.  20962 

.97778 

.  22665 

•97398 

.24362 

.96987 

54 

7 

•17565 

.98445 

.  9281 

.98  24 

.20990 

.97772 

.22693 

•97391 

.24390 

.96980 

53 

8 

•17594 

.98440 

•  9309 

.  1019 

.97766 

.22722 

.97384 

.24418 

•96973 

52 

0 

.17623 

•98435 

•  9338 

!98  12 

•  i°47 

.97760 

.22750 

•97378 

.24446 

.96966 

.17651 

.98430 

.  9366 

.98  07 

•  I076 

•97754 

.22778 

•97371 

.24474 

.96959 

5" 

, 

.17680 

.98425 

•  9395 

.98101 

.  1104 

.97748 

.228O7 

•97365 

•24503 

.96952 

4" 

12 

.17708 

.98420 

•  9423 

.98096 

•  "32 

•97742 

.22835 

•97358 

•24531 

.96945 

48 

13 

.  17737 

.98414 

•  9452 

.98090 

.  1161 

•97735 

.22863 

•97351 

•24559 

.96037 

47 

'4 

.  17766 

.98409 

.  9481 

.98084 

.  1189 

.97729 

.22892 

•97345 

•24587 

•96§3o 

46 

5 

•'7794 

.98404 

.  9509 

.98079 

,  1218 

•97723 

.22920 

•97338 

.24615 

•96923 

45 

6 

.17823 

.98399 

•  9538 

.98073 

.  1246 

•97717 

.22948 

•97331 

.  24644 

.96916 

44 

7 

.17852 

.98394 

.  9566 

.98067 

•  1275 

•9771.1 

.22977 

•973*5- 

.23672 

.96909 

4i 

8 

.17880 

.98389 

•  9595 

.98061 

•  1303 

•97705 

•23005 

•97318 

754700 

.96902 

42 

9 

.17909 

•98383 

.  9623 

.98056 

•  '33' 

.97698 

•23033 

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.24728 

.96894 

4' 

20 

•T7937 

-98378 

.  9652 

.98050 

•  '360 

.97692 

.23062 

•97304 

•24756 

.96887 

4" 

21 

.17966 

•98373 

.  9680 

.98044 

1388 

.97686 

.23090 

.97298 

.24784 

.96880 

30 

22 

•'7995 

.98368 

•  97°9 

.98039 

•  '4'7 

.97680 

.23118 

.97291 

.24813 

.96873 

23 

24 

.18023 
.18052 

.98362 
•98357 

•  9737 
.  9766 

,98033 
.98027 

•  '445 
•  '474 

•97673 
.97667 

•23146 
•23175 

•97284 
.97278 

.24841 
.24869 

.96866 
.96858 

37 

25 

.  18081 

•98352 

•  9794 

.98021 

•  1502 

.97661 

.23203 

•97271 

.24897 

.96851 

35 

26 

.18109 

•98347 

•  9823 

.98016 

•  1530 

•97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18138 

•98341 

.  9851 

.98010 

•  '559 

.97648 

.23260 

•97257 

•24954 

.96837 

.18166 

•98336 

.  9880 

.98004 

•  1587 

.97642 

.23288 

•97251 

.24982 

.96829 

3* 

99 

.  18195 

•9833' 

.  9908 

.97998 

.  1616 

.97636 

.23316 

•97244 

.25010 

.96822 

31 

JO 

.18224 

•98325 

•  9937 

.97992 

.  1644 

.97630 

•23345 

•97237 

.25038 

.96815 

30 

.31 

.18252 
.18281 

.98320 

.98315 

•  9965 

•  97987 
.07981 

.  1672 

.97623 

•23373 

•97  3° 

.25066 

.96807 
.96800 

M 

33 

.18309 

.98310 

!  20022 

•97975 

•  '729 

.97611 

•23429 

•97  '7 

.25122 

•96793 

27 

34 

•'8338 

.98304 

.20051 

•97969 

•  '758 

.97604 

.23458 

•97  10 

•25151 

.96786 

P 

.18367 
•18395 

.98299 
.08294 

.20079 
.20108 

.97963 
•97958 

.  1786 
.  1814 

-97598 
•97592 

.23486 
•23514 

•97  °3 
•97  Q6 

•25'  79 
.25207 

•96778 
.96771 

25 

37 

.18424 

.98288 

.20136 

•97952 

•  1843 

•97585 

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•  97  89 

•25235 

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•M 

3B 

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.  1871 

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•97  82 

•25263 

.96756 

30 
40 

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.  1899 
.  1928 

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•97  76 
•97  69 

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to 

4' 

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•  1956 

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.97  62 

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HJ 

2 

•  18567 

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.97922 

.  1985 

•97553 

•23684 

•97  55 

•25376 

.96727 

18 

i 

-18595 

.98256 

•  20307 

.97916 

•  2013 

•97547 

.23712 

•97  48 

.25404 

•96719 

17 

4 

.18624 

.98250 

•20336 

•97910 

.  2041 

•97541 

•23740 

•97  4' 

•25432 

.96712 

6 

5 

.18652 

.98245 

.20364 

•97905 

.  2070 

•97534 

.23769 

•97  34 

.25460 

.96705 

5 

6 

.18681 

.98240 

•20393 

.97899 

.  2098 

•97528 

.23797 

•  97  27 

.25488 

.96697 

4 

7 

.18710 

•  98234 

.2O421 

•97893 

.  2126 

•97521 

.23825 

•  97  20 

•25516 

.96690 

3 

8 

.18738 

.98229 

.  20450 

.97887 

•  ^155 

•97515 

•23853 

•97  13 

•25545 

.96682 

2 

49 

.18767 

.98223 

.20478 

.9788, 

•  2183 

.97508 

.23882 

.97  06 

•25573 

.96675 

I 

5" 

•  18795 

.98218 

.20507 

•97875 

.  2212 

.97502 

.23910 

.97  oo 

.25601 

.96667 

1 

Si 

.18824 

.98212 

•20535 

.97869 

.  2240 

•97  96 

•23938 

.97093 

.25629 

.96660 

g 

5-' 

18852 

.98207 

•20563 

.97863 

.  2268 

•97  89 

.23966 

.97086 

•25657 

.96653 

8 

53 

.18881 

.98201 

•  20592  f 

•97857 

•  2297 

•97  83 

•23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98  96 

.  2O62O 

•97851 

•  2325 

•  97  76 

.24023 

.97072 

•25713 

.96638 

6 

I 

•18938 

!'i 

.20649 
.20677 

•97845 
•97839 
•97833 

•  2353 
.  2382 

•97  70 
•97463 

.24051 
.24079 

.97065 
.97058 

•25741 
.25769 

.96630 

:?i 

P 

19024 

•  98  74 

•20734 

•97827 

•  2438 

•97450 

•24136 

.97044 

.25826 

59 

.19052 

.98  68 

•  20763 

.97821 

.  2467 

•97*44' 

.24164 

•97037 

•25854 

!  96600 

Go 

.19081 

.98  63 

.2079! 

•97815 

•  249=; 

<2Q7437 

.24192 

.97030 

.25882 

•96593 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

' 

p 

7 

9° 

7< 

J° 

7 

7° 

7 

5° 

7 

.0 

) 

26 


NATURAL  SINES  AND  COSINES. 


I 

5° 

I 

5° 

i 

1" 

I 

8° 

I 

9° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

.25882 
.25910 

•96593 
•96585 

•27564 
.27592 

.96,26 
.961,8 

.29237 

.95630 
.95622 

.30902 
.30929 

.95106 
.95097 

•32557 
•32584 

•94552 
•94542 

60 

=  9 

2 

•25938 

•96578 

.27620 

.961,0 

•29293 

•95613 

•30957 

.95088 

.32612 

•94533 

3 

•25966 

.96570 

.27648 

.96102 

.2932, 

•95605 

.30985 

•95079 

.32639 

•94523 

57 

4 
5 

.26022 

.96562 
•96555 

.27676 
.27704 

.96094 
.96086 

•29348 
•29376 

•95596 
.95588 

.31012 
•3*040 

.95070 
.95061 

•32694 

•945M 
.  94504 

56 

=  5 

6 

.  26050 

•96547 

.27731 

.96078 

.29404 

•95579 

.3,068 

.95052 

•32722 

•94495 

54 

7 

.26079 

.96540 

•27759 

.96070 

.29432 

•9557* 

•3*095 

•95043 

•32749 

.94485 

53 

8 

.26107 

•96532 

.27787 

.96062 

.29460 

.95562 

•95033 

•32777 

.94476 

52 

9 

•26135 

.96524 

.27815 

.96054 

.29487 

•95554 

.31151 

.95024 

•32804 

.944^6 

51 

10 

.26163 

•96517 

•27843 

.96046 

•29515 

•95545 

.3,178 

•95015 

•32832 

•94457 

50 

,1 

.26191 

.96509 

.27871 

.96037 

•29543 

•95536 

.31206 

.95006 

.32859 

•94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

•29571 

.95528 

•3*233 

•94997 

•32887 

.94438 

'i 

.26247 

.96494 

.27927 

.96021 

•29599 

•95519 

.3,26, 

.94988 

.32914 

.94428 

47 

'4 

.26275 

.96486 

•27955 

.96013 

.29626 

•955** 

.3*289 

•94979 

.32942 

.94418 

46 

'5 

.26303 

•96479 

.27983 

.96005 

•  29654 

•95502 

•3*3*6 

.94970 

•32969 

.94409 

45 

16 

•26331 

.96471 

.28011 

•95997 

.29682 

•95493 

•3*344 

.94061 

•32997 

•94399 

44 

17 

•26359 

.96463 

•28039 

•95989 

.29710 

•95485 

•3*372 

•94952 

•33024 

.94390 

43 

18 

.26387 

•96456 

.28067 

.95981 

•29737 

•95476 

•94943 

•33°5* 

.94380 

42 

*9 

.264,5 

.96448 

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•95972 

.29765 

•95467 

•3*427 

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•94370 

20 

.26443 

.96440 

.28123 

•95964 

.29793 

•95459 

•3*454 

•94924 

•33*o6 

.94361 

40 

21 

.2647* 

•96433 

.28150 

•95956 

.29821 

•95450 

•3*482 

•949*5 

•33*34 

•94351 

39 

22 

.26500 

.96425 

.28178 

.95948 

.29849 

•9544* 

.31510 

.94906 

•33*6* 

.94342 

38 

•3 

.26528 

•96417 

.28206 

.95940 

.29876 

•95433 

•3*537 

•94897 

•33189- 

•94332 

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.26556 

.96410 

.28234 

•9593* 

.29904 

•95424 

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.94888 

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•94322 

3" 

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.96402 

.  28262 

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2*1 

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.96394 

.28290 

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•95407 

.3l62O 

.94869 

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•94303 

34 

27 

.26640 

.96386 

.283,8 

•95907 

.29987 

•95398 

.31648 

.94860 

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.94293 

33 

2S 

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.26668 
.26696 

.96379 
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•95898 
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•95389 
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.94851 
.94842 

•33326 

•33353 

.94284 
•94274 

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31 

3" 

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.96363 

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.95882 

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•95372 

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•94832 

•3338* 

.94264 

3° 

31 

.26752 

•96355 

.28429 

.95874 

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3*758 

.94823 

.33408 

94254 

29 

3* 

.26780 

•96347 

•28457 

.95865 

.30126 

•95354 

i  31786 

.94814 

•33436 

•94245 

28 

J3 

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.96340 

•28485 

•95857 

•30*54 

•95345 

•3*8*3 

.94805 

•33463 

•94235 

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34 

.26836 

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.95849 

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•94795 

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35 

.26864 

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•94215 

25 

5<S 

.26V 

.963*6 

.28569 

.95832 

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•953*9 

.31896 

•94777 

•33545 

.94206 

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.26020 

.96308 

•28597 

.95824 

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.94196 

2; 

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•94758 

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.94186 

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.26976 

•96293 

.28652 

.95807 

.30320 

•95293 

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•94749 

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•94176 

21 

4" 

.27004 

.96285 

.28680 

•95799 

•30348 

•95284 

.32006 

.94740 

•33655 

.94167 

20 

4' 
42 

•27032 
.27060 

.96277 
.96269 

.28708 
.28736 

•9579* 

.95782 

•30376 
•30403 

•95275 
..95266 

•32034 
.32061 

.94730 
.94721 

.33682 
•337*0 

•94*57 
•94147 

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.27088 

.96261 

.28764 

•95774 

•3°43* 

•95257 

.32089 

•947*2 

•33737 

•94137 

7 

44 

.27116 

•96253 

.28792 

•95766 

•30459 

.95248 

.32116 

.94702 

•  33764 

.94127 

6 

45 

•27*44 

.96246 

.28820 

•95757 

.30486 

.95240 

.32144 

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.94118 

5 

4'' 

.27172 

.96238 

.28847 

•95749 

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•95231 

.32171 

.94684 

.94108 

4 

47 

.27200 

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•  30542 

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.94674 

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i 

4K 
49 

.27228 
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.96222 
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.94088 
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5o 

.27284 

.96206 

•28959 

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0 

s* 

•273*2 

.96198 

.28987 

•95707 

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•95*86 

.32309 

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.96190 

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•95698 

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•95177 

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.94627 

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8 

53 

•27368 

.96,82 

.29042 

.95690 

.30708 

•95168 

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.  3401  i 

•94039 

7! 

54 

.27396 

.96,74 

.29070 

.95681 

•30736 

•95*59 

•32392 

.94609 

•34038 

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6 

55 

.27424 

.96166 

.29098 

•95673 

•30763 

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•324*9 

•94599 

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5 

5" 

•27452 

.96158 

.29126 

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.95142 

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4  1 

57 

.27480 

•96150 

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.95656 

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.93999 

3 

58 

.27508 

.96*42 

.29182 

•95647 

.30846 

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.32502 

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5') 

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•96134 

.29209 

•95639 

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.94561 

•34*75 

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1  1 

§0 

•27564 

.96126 

•29237 

•95630 

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•95106 

•32557 

•94552 

.34202 

•93969 

."-\ 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

7] 

7 

4° 

7 

5° 

7 

2° 

7 

1° 

7 

D° 

NATURAL  SINES  AND  COSINES. 


2 

D° 

2 

i° 

2 

j° 

2 

5°   ! 

2, 

r 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

.34202 

.93969 

•35837 

•93358 

•3746. 

.927.8 

•39073 

.92050 

•40674 

•9  355 

60 

I 

.34229 

•93959 

.35864 

•93348 

.37488 

•92707 

.39100 

.92039 

.40700 

•9  343 

59 

2 

•34257 

•93949 

•35891 

•93337 

•37515 

.92697 

•39127 

.92028 

•40727 

•9  331 

3 

.34284 

•93939 

•35918 

•93327 

•37542 

.92686 

•39153 

.92016 

•40753 

•9  3i9 

57 

•343" 

•93929 

•35945 

•933i6 

•37569 

.92675 

.39.80 

.92005 

.40780 

•9  307 

S6 

5 

•34339 

.939.9 

•35973 

•93306 

•37595 

.92664 

.39207 

•91994 

.40806 

•9  295 

55 

6 

.34366 

•  03909 

.36000 

•93295 

.37622 

.92653 

•39234 

.91982 

.40833 

•9  283 

54 

7 

•  34393 

.93899 

.36027 

•93285 

.37649 

.92642 

.39260 

.91971 

.40860 

•9  272 

53 

S 

•34421 

.93889 

•36054 

•93274 

.37676 

•92631 

.39287 

•91959 

.40886 

.9  260 

52 

9 

.34448 

•93879 

.3608. 

•93264 

•37703 

.92620 

•39314 

.91948 

.40913 

•9  48 

51 

10 

•34475 

.93869 

.36.08 

•93253 

•37730 

.92609 

•39341 

•91936 

•40939 

•9  36 

50 

„ 

•34503 

•93859 

•36i35 

•93243 

•37757 

.92598 

•39367 

•91925 

.40966 

•9  24 

44 

12 

•34530 

.93849 

.36.62 

•93232 

•37784 

.92587 

•39394 

•91914 

.40992 

.9  .2 

48 

13 

•34557 

•93839 

•36.90 

.93222 

•37811 

•92576 

.3942. 

.91902 

.9  oo 

47 

•34584 

.93829 

•36217 

.932.1 

•37838 

•92565 

•39448 

.91891 

•  1045 

.9  88 

4" 

15 

•34612 

.938.9 

•36244 

•9320. 

•37865 

•92554 

•39474 

.9.879 

.  .072 

•9  76 

45 

16 

•34639 

.93809 

.36271 

.93.90 

.37892 

•92543 

•39501 

.9.868 

.  ,098 

•9  64 

44 

17 

.34666 

•93799 

.36298 

.93180 

•37919 

.92532 

•39528 

•91856 

.  1125 

•9  52 

43 

iS 

.34694 

•93789 

•36325 

.93.69 

•37946 

.9252. 

•39555 

.9,845 

•41.51 

•9  40 

42 

ig 

.34721 

•93779 

•36352 

.93.59 

•37973 

.925.0 

•3958i 

•91833 

.41178 

.9  28 

41 

BO 

•34748 

.93769 

•36379 

.93148 

•37999 

•92499 

.39608 

.9,822 

.41204 

.9  .6 

4" 

21 

•34775 

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.36406 

•93137 

.38026 

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39 

22 

.34803 

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.91799 

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.9  092 

J8 

2-i 

.34830 

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.93.16 

.38080 

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.39688 

.9.787 

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37 

24 

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.4.3.0 

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.34884 

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.9.764 

.9  056 

35 

26 

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.93084 

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34 

27 

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33 

28 

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.4.4.6 

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p 

•34993 

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34 

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.9,660 

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26 

35 

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36 

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it; 

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i  •; 

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15 

4" 

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•35647 

•93431 

.37272 

.92794 

.38886 

.92130 

.40488 

•9M37 

•42077 

.90717 

7 

54 

•35674 

.93420 

•37209 

•92784 

•38912 

.92119 

.405.4 

•91425 

.42.04 

.90704 

6 

55 

•35701 

.93410 

•37326 

•92773 

•38939 

.92107 

•40541 

.91414 

.42130 

.90692 

5 

5" 

•35728 

.93400 

•37353 

.92762 

.38966 

.92096 

.40567 

.91402 

.42156 

.90680 

4 

57 

•35755 

•93389 

.37380 

.9275. 

.38993 

.92085 

•4°594 

.91390 

.42.83 

.90668 

3 

SS 

•35782 

•93379 

•  37407 

.92740 

.39020 

.92073 

.40621 

•91378 

.42209 

•90655 

•J 

59 
So 

.358.0 
•35837 

•93368 
•93358 

•37434 
•37461 

•92729 
.927,8 

.39046 
•  39°73 

.92062 
.92050 

.40647 
•40674 

.91366 
•91355 

•42235 
.42262 

•90643 
.9063. 

I 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

6 

9° 

6 

3° 

6 

7° 

6 

6° 

6 

5 

28 


NATURAL  SINES  AND  COSINES. 


2C 

21 

2 

1° 

2 

2( 

)° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

I 

.42262 
.  2288 

.00631 
.90618 

•43837 
•43863 

.89879 
.89867 

•45399 
•45425 

.89101  i 
.89087 

•46947 
•46973 

.88295  I 

.4848! 
.48506 

.87462 
.87448 

» 

•  2315 

.90606 

.43889 

•89854 
89841 

•4545' 

.80074 
.89061 

.46999 

!  88267 
88254 

.48532 

•87434 

4 

•  2367 

190582 

•43942 

.89828 

•45503 

.89048 

.47050 

.88240 

.48583 

.87406 

•  2394 

.90569 

.43968 

.89816 

•45529 

.89035 

.47076 

.88226 

.48608 

•87391 

6 

.  2420 

•9°557 

.  43994 

.89803 

•45554 

.89021 

.47101 

.88213 

•48634 

•87377 

7 

.  2446 

•9°545 

.44020 

.89790 

.45580 

.89008 

.47127 

•48659 

•87363 

8 

•  2473 

.90532 

.44046 

.89777 

.45606 

.88995 

•47153 

.88185 

.48684 

•87349 

9 

•  2499 

.90520 

•44072 

.89764 

•45632 

.88981 

.47178 

.48710 

•87335 

'  252l 

.90507 

.44098 

•89752 

•45658 

.88968 

•47204 

'.88158 

•48735 

.87321 

it 

•  2552 

.90495 

.44124 

.89739 

.45684 

•88955 

.47229 

.88144 

.4876! 

.87306 

12 

.  2578 

.90483 

•44I5I 

.89726 

•45710 

.88942 

•47255 

.88130 

.48786 

.87292 

:3 

.42604 

.90470 

•44!77 

.89713 

•45736 

.88028 

.47281 

.48811 

.87278 

'4 

.42631 

•90458 

.44203 

.89700 

•45762 

.88915 

.47306 

'.88103 

.48837 

.87264 

Jj 

.42657 

.  90446 

•44229 

.89687 

•45787 

.88902 

•47332 

.48862 

.87250 

10 

.42683 

•90433 

•44255 

.89674 

•458i3 

.88888 

.47358 

.88075 

.48888 

•87235 

3 

.42709 
.42736 

.90421 
.90408 

.44281 
•44307 

.89662 
.89649 

•45839 
.45865 

.88875 

•47383 
.47409 

.88062 
.88048 

.48913 

.48938 

.87221 
.87207 

19 

.42762 

.90396 

•44333 

.89636 

•45891 

.88848 

•47434 

.88034 

.48964 

•87193 

20 

.42788 

.90383 

•44359 

•89623 

•459^7 

.88835 

.47460 

.88020 

.48989 

.87178 

21 

.  28I5 

•90371 

•44385 

.89610 

•45942 

.88822 

.47486 

.88006 

.49014 

.87164 

22 

.  2841 

•90358 

.44411 

•89597 

.45968 

.  88808 

•475" 

•87993 

.49040 

.87:50 

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.  2867 

.90346 

•44437 

.89584 

•45994 

.88795 

•47537 

.87979 

.49065 

•87136 

24 

.  2894 

•90334 

.44464 

•89571 

.46020 

.88782 

•47562 

.87965 

.49090 

.87121 

25 

2920 

.90321 

.  44490 

•89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

26 

.  2946 

.00309 

.44516 

•  89545 

.46072 

•88755 

.47614 

•87937 

.49141 

.87093 

27 

•42972 

.90296 

•44542 

.89532 

.46097 

.88741 

•47639 

.87923 

.49166 

.87079 

28 

.42999 

.90284 

.44568 

.89519 

•46123 

.88728 

•47665 

.49192 

.87064 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.887.5 

.47690 

.87896 

.49217 

-87050 

3" 

•43051 

.90259 

.44620 

.89493 

•46175 

.88701 

.47716 

.87882 

.49242 

.87036 

3i 

•43077 

.  90246 

.44646 

.89480 

.46201 

88688 

•47741 

.87868 

.49268 

.87021 

32 

.43104 

.90233 

•44672 

.89467 

.46226 

.88674 

•47767 

.87854 

•49293 

.87007 

33 

•43T3° 

.90221 

.44698 

.89454 

.46252 

.88661 

•47793 

.87840 

.49318 

.86993 

34 

•43!56 

.  90208 

•44724 

.89441 

.46278 

.88647 

.47818 

.87826 

•49344 

.86978 

3 

•43182 
.43209 

.90196 
.90183 

.  44750 
.44776 

.89428 
.89415 

.46304 
•46330 

.88634 
.88620 

•47844 
.47869 

.87812 
.87798 

•49369 
•49394 

.86964 
.86949 

37 

•43235 

.90171 

.44802 

.89402 

•46355 

.88607 

•47895 

.87784 

.49419 

•86935 

38 
39 

•43261 
.43287 

.90158 
.90146 

.44828 
.44854 

•89389 
.89376 

•46381 
.46407 

•88593 
.88580 

.47920 

•47946 

.87770 
•87756 

•49445 

.40470 

.86921 
.86906 

40 

•433'3 

.90133 

.44880 

.89363 

•46433 

.88566 

•47971 

•87743 

•49495 

.86892 

41 

•4334° 

.90120 

.44906 

.89350 

•46458 

•88553 

•47997 

.87729 

•49521 

.86878 

42 

•433«6 

.90108 

•44932 

•89337 

.46484 

.88539 

.48022 

•87715 

.49546 

.86863 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

•4957' 

.86849 

44 

.43418 

.93082 

.44984 

.893:1 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

45 

•43445 

.90070 

.45010 

.89298 

•46561 

.88499 

.48099 

•87673 

.49622 

.86820 

46 

•43471 

.90057 

•45036 

.89285 

•46587 

.88485 

.48124 

•87659 

.49647 

.86805 

47 

•  43497 

.90045 

.45062 

.89272 

•46613 

.88472 

.48150 

.87645 

.49672 

.86791 

48 

•43523 

.90032 

.45088 

.89259 

•46639 

.88458 

•48i75 

.87631 

•49697 

.86777 

49 

•43549 

.90019 

•45H4 

.89245 

.46664 

.88445 

.48201 

.876i7 

•49723 

.86762 

5'-' 

•43575 

.90007 

.45140 

.89232 

.46690 

.87603 

.49748 

.86748 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

•49773 

.86733 

53 

.43628 

.89981 

•45J92 

.89206 

.46742 

.88404 

•48277 

•87575 

.49798 

.86719 

53 

.43654 

.89968 

•452X8 

.89193 

•46767 

.88390 

•48303 

.87561 

.49824 

.86704 

54 

.43680 

.89956 

•45243 

.89  80 

•46793 

•88377 

.48328 

.87546 

.49849 

.86690 

55 

.43706 

.89943 

.45269 

.89  67 

.46819 

.88363 

•48354 

•87532 

.49874 

.86675 

56 

•43733 

.89930 

•45295 

.89  53 

.46844 

.88349 

•48379 

.87518 

.49899 

.86661 

57 

•43759 

.89918 

•45321 

.8940 

.46870 

•88336 

•48405 

.87504 

.49924 

.86646 

58 

•43785 

.89905 

•45347 

.89  27 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

59 

.43811 

.89892 

•  45373 

.89  14 

.46921 

.88308 

•48456 

.87476 

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fo 

^•438?7 

.89879 

•45399 

.890, 

•46947 

.88295 

.48481 

.87462 

.50000 

.86603' 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

L 

6 

4° 

6 

3° 

6 

2° 

6 

1° 

6 

O° 

NATURAL  SINES  AND  COSINES. 


3< 

) 

31 

32 

0 

32 

0 

2>A 

0 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

.50000 
.50025 

.86603 
.86588 

.51504 
•51529 

•85717 
•85702 

•  52992 
•53017 

.84805 
.84789 

•54464 
.54488 

.83867 
.8385, 

•55919 
•55943 

.82904 
.82887 

So 

M 

.50050 

.86573 

•51554 

•85687 

•53041 

.84774 

•545^3 

.83835 

•55968 

.82871 

.50076 

.86559 

•51579 

.85672 

.53066 

•84759 

•54537 

.83819 

•55992 

.82855 

57 

.50101 

-86544 

•85657 

•53091 

•84743 

•5456i 

.83804 

.56016 

.82839 

56 

.50126 

.86530 

.51628 

.85642 

•53"5 

.84728 

•54586 

.83788 

.56040 

.82822 

55 

.50151 

.86515 

•51653 

.85627 

•53I40 

.84712 

.54610 

•83772 

.56064 

.82806 

54 

.50176 

.86501 

.51678 

.85612 

•53164 

.84697 

•54635 

•83756 

.56088 

.82790 

S3 

.50201 

.86486 

•51703 

•85597 

.84681 

•54659 

.83740 

.56112 

•82773 

52 

.50227 

.86471 

.51728 

.85582 

•53214 

.84666 

•54683 

.83724 

•56136 

•82757 

SI 

I 

.50252 

•86457 

•85567 

•53238 

.84650 

.54708 

•83708 

.56160 

.82741 

3" 

12 

13 

•50277 
.50302 
.50327 

.86442 
.86427 
.86413 

•51778 
.51803 
.51828 

•85551 

•85536 
.85521 

•53263 
.53288 
•53312 

.8463 
.8461 
.8460 

•54732 
•54756 
•5478i 

.83692 
•83676 
.83660 

.56184 
.56208 
•  56232 

.82724 
.82708 
.82692 

a 

47 

•50352 

.86398 

.51852 

.85506 

•53337 

.8458 

.54805 

•83645 

.56256 

•82675 

46 

15 

•50377 

.86384 

•51877 

.85491 

•53361 

•  8457 

•54829 

.83629 

.56280 

.82659 

45 

16 

.50403 

.86369 

•  5  1902 

.85476 

.53386 

•  8455 

•54854 

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.56305 

.82643 

44 

17 

.50428 

.86354 

.51927 

.85461 

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•  8454 

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•83597 

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43 

18 

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19 

.50478 

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20 

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.85416 

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23 

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!  86266 

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Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

5 

9° 

5 

3° 

5 

7° 

5 

5° 

5 

5° 

30 


NATURAL  SINES  AND  COSINES. 


3J 

o 

3< 

;° 

3' 

1° 

3< 

31 

) 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

I 

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.81915 
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.80902 
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.60.82 
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57 

4 

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.6.658 

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56 

5 

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55 

6 
7 

.57501 
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.  58920 
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.6032. 

•60344 

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.61704 
.61726 

.78694 
.78676 

.63068 
.63000 

•77605 
.77586 

54 
53 

8 

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.60367 

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.6.749 

.78658 

.63..3 

•77568 

52 

9 

•57572 
•57596 

IIS 

.58090 
.59014 

.80748 
.80730 

.60390 
.60414 

.  79706 
.79688 

.6.772 
•6.795 

.78640 
.78622 

•63.35 
.63158 

•77550 
•77531 

5° 

„ 

•57619 

.81731 

•59037 

.80713 

.60437 

•79671 

.6.8.8 

.78604 

.63180 

•77513 

49 

12 

•  57643 

.5906. 

.80696 

.60460 

•79653 

.6.841 

.78586 

.63203 

•77494 

48 

13 
'4 

'•57691 

]8i68i 

.59084 
.59108 

.80679 
.80662 

.60483 
.60506 

•79635 
.796.8 

.6.864 
.6.887 

•78568 
•78550 

.63225 
.63248 

•77476 
•77458 

47 

46 

'5 

•57715 

.81664 

•59I31 

.80644 

.60529 

.79600 

.6.909 

•78532 

•6327. 

•77439 

45 

1  6 

•57738 

.81647 

•59154 

.80627 

•60553 

•79583 

.6.932 

•78514 

.63293 

.7742. 

44 

17 

•57762 

.81631 

•59178 

.806.0 

.60576 

•79565 

•6.955 

.78496 

.633.6 

•  77402 

43 

18 

•57786 

.81614 

.  59201 

.80593 

.60599 

•79547 

.6.978 

•78478 

•63338 

•77384 

42 

19 

•57810 

•81597 

•59225 

.80576 

.60622 

•79530 

.62001 

.78460 

.6336. 

.77366 

41 

20 

•57833 

•59248 

.80558 

.60645 

.79512 

.62024 

.78442 

•63383 

•77347 

4° 

22 

•57857 
.57881 

•81563 
.81546 

.59272 
•59295 

.8054. 
.80524 

.60668 
.6069. 

•79494 
•  79477 

.62046 
.62069 

.78424 
.78405 

.63406 

.63428 

•77329 
.773.0 

P 

23 

•57904 

.81530 

•593l8 

.80507 

.607.4 

•  79459 

.62092 

•78387 

•63451 

.77292 

37 

24 

•57928 

.81513 

•59342 

.80489 

.60738 

•79441 

.621.5 

•78369 

•63473 

•77273 

36 

26 

•57952 
•57976 

.81496 
.81479 

•59365 
•59389 

.80472 
.80455 

!  60784 

•  79424 
.79406 

.62.38 
.62.60 

•7835' 
•78333 

•63496 
-63518 

•77255 
.77236 

35 
34 

27 

•57999 

.81462 

.594.2 

.80438 

.60807 

.79388 

.62.83 

•78315 

•63540 

.772.8 

33 

28 

.58023 

.81445 

•59436 

.80420 

.60830 

•79371 

.62206 

.78297 

•63563 

.77.99 

32 

'2() 

.58047 

.81428 

•59459 

.80403 

.60853 

•79353 

.62229 

.78279 

•63585 

•77.8. 

3' 

30 

.58070 

.81412 

.59482 

.80386 

.60876 

•79335 

.62251 

.7826. 

.63608 

.77.62 

30 

31 

.58094 

•8i395 

-  595°6 

.80368 

.60899 

•793i8 

.62274 

.78243 

.63630 

•77J44 

29 

.58118 

•81378 

•59529 

.8035. 

.60922 

•79300 

.62297 

•78225 

•63653 

.77.25 

28 

33 
34 

.58141 
.58165 

.81361 
.81344 

•59552 
•  59576 

•80334 
.803.6 

.60945 
.60968 

-.79282 
.79264 

.62320 
.62342 

.78206 

.63675 
.63698 

.77107 
.77088 

11 

35 

.58189 

•81327 

•59599 

.  80299 

.6099. 

•  79247 

.62365 

.78170 

.63720 

.  77070 

25 

36 
37 

•58212 
.58236 

.8.310 
.81293 

•59622 
.59646 

:  80264 

.6.0.5 
.6.038 

.79229 
.7921. 

.'624.1 

.78.52 
•78.34 

-63742 
•63765 

.77051 
•77033 

24 
23 

38 

.58260 

•59669 

.80247 

.6.06. 

•  79*93 

•62433 

.781.6 

.63787 

.770.4 

22 

39 

•58283 

.81259 

•  59693 

.80230 

.6.084 

.79.76 

.62456 

.78098 

.638.0 

.76996 

21 

40 

•58307 

.81242 

.59716 

.802.2 

.6.107 

•62479 

•78079 

.63832 

.76977 

20 

41 
42 

43 

•58330 

•58354 
•58378 

li 

•59739 
•  59763 
.59786 

.80.95 
.80  78 
.80  60 

.6.130 
.61.53 
.6.176 

.79.40 

.79.22 

.62502 
.62524 
•62547 

.78061 
•78043 
•  78025 

.63854 
.63877 
.63899 

•76959 
.76940 
.76921 

'9 
17 

44 

.58401 

.81174 

.59809 

.80  43 

.6.199 

•79087 

•62570 

.78007 

.63922 

6 

45 

•58425 

.81157 

.59832 

.80  25 

.6.222 

.79069 

.62592 

.77988 

.63944 

.76884 

5 

46 

.58449 

.81140 

•59856 

.80  08 

.6.245 

•79051 

.62615 

•  77970 

.63966 

.  76866 

4 

47 

.58472 

.81123 

•59879 

.8009. 

.6.268 

•79033 

.62638 

•77952 

.63989 

•76847 

3 

48 

.58496 

.8no6 

.59902 

.80073 

.6.29. 

.79016 

.62660 

•77934 

.643.1 

.76828 

2 

49 

•58519 

.81089 

.59926 

.80056 

.6.3.4 

.78998 

.62683 

.77916 

•64033 

.768.0 

I 

5" 

•58543 

.81072 

•59949 

.80038 

•6.337 

.78980 

.62706 

.77897 

.64056 

.76791 

.0 

s, 

•58567 

.81055 

.59972 

.8002. 

.6.360 

.78962 

.62728 

•77879 

.64078 

.76772 

9 

.52 

•58590 

.8.038 

•59995 

.80003 

•6.383 

.78944 

.6275. 

.7786. 

.64100 

•76754 

8 

53 

•58614 

.8.O2T 

.60019 

.79986 

.6.406 

.78926 

•62774 

•77843 

.64.23 

•76735 

7 

54 

•58637 

.8.004 

.60042 

.79968 

.61429 

.78908 

.62796 

•77824 

.64145 

.76717 

6 

55 

.58661 

.80987 

.60065 

•79951 

.6.45. 

.7889. 

.62819 

.77806 

.64167 

.76698 

5 

56 

.58684 

.80970 

.60089 

•79934 

•61474 

•78873 

.62842 

.77788 

.64190 

•76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

•6.497 

•78855 

.62864 

.77769 

.642.2 

.76661 

3 

58 

•58731 

.80936 

.60.35 

•79899 

.6.520 

.78837 

.62887 

•77751 

.64234 

.76642 

2 

59 

•58755 

.80919 

.60158 

.7988. 

•6.543 

.788.9 

.62909 

•77733 

•64256 

.76623 

I 

6,, 

•58779 

.80902 

.60.82 

.79864 

.6.566 

.7880, 

.62932 

•77715 

.64279 

.76604 

0 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

5' 

\° 

5. 

J° 

5 

1° 

5 

[° 

5C 

)° 

NATURAL  SINES  AND  COSINES. 


31 


4< 

>° 

4 

i° 

4 

2° 

4 

3° 

4< 

^° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

•64279 
•64301 

.76604 
.76586 

.65606 

.65628 

•75471 
•75452 

.66913 
•66935 

•  43M 

•  4295 

.68200 
.68221 

•73135 
.73116 

.69466 
.69487 

•  1934 

60 
59 

•64323 

•76567 

•65650 

•75433 

.66956 

•  4276 

.68242 

.73096 

.69508 

1894 

58 

.64346 
.64368 

•76548 
•7653° 

.65672 
.65694 

•754I4 
•73395 

.66978 
.66999 

•  4256 

•  4237 

.68264 
.68285 

•  73076 
•  73°56 

•69529 
•69549 

•  1873 
•  1853 

9 

•64390 

.76511 

.65716 

•75375 

.67021 

•  4217 

.68306 

•  73036 

.69570 

•  1833 

55 

.64412 

.  76492 

•65738 

•75356 

•67043 

•68327 

.73016 

.69591 

.  1813 

54 

•64435 

•76473 

•65759 

•75337 

.67064 

•  4T78 

•68349 

.72996 

.69612 

•  1792 

53 

•64457 

•76455 

•65781 

.67086 

•  4J59 

.68370 

•72976 

.69633 

•  '772 

52 

.64479 

.76436 

.65803 

•75299 

.67107 

•  4139 

.68391 

•72957 

•69654 

•  1752 

1 

.64501 

.76417 

•65825 

.75280 

.67129 

.68412 

•72937 

•69675 

•  1732 

So 

, 

.64524 

.76398 

.65847 

•75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.  1711 

49 

T 

.64546 

.76380 

.65869 

.75241 

.67172 

.74080 

.68455 

•  72897 

.69717 

.  1691 

48 

T 

.64568 

•7636! 

.65891 

.75222 

.67194 

.74061 

.68476 

•  2877 

•69737 

.  1671 

47 

J 

.64590 

.76342 

•  75203 

.67215 

•  74041 

.68497 

•  2857 

.69758 

•  1650 

46 

I 

.64612 

•76323 

•65935 

•75184 

•67237 

.74022 

.68518 

•  2837 

.69779 

•  ^630 

45 

16 
17 

.64635 
•64657 

•  76304 
.76286 

•65956 
.65978 

•75»5 

•75M6 

•67258 
.67280 

.74002 
•73983 

•68539 
.68561 

.  2817 
•  2797 

.69800 
.69821 

•  1590 

44 
43 

1  8 

.64679 

.76267 

.66000 

.75126 

•  67301 

•73963 

.68582 

•  2777 

.69842 

•  1569 

42 

19 

.64701 

.76248 

.66022 

.75107 

•67323 

•73944 

.68603 

.72757 

.69862 

•  '549 

41 

20 

•64723 

.76229 

.66044 

.75088 

•67344 

•73924 

.68624 

•72737 

.69883 

•  1529 

40 

21 

.64746 

.76210 

.66066 

.75069 

•67366 

.73904 

.68645 

•72717 

.69904 

•  1508 

39 

22 

.64768 

.76192 

.66088 

•  75050 

•67387 

•73885 

.68666 

•72697 

•69925 

.  1488 

38 

23 

.64790 

•76173 

.66109 

•  7503° 

.67409 

.73865 

.68688 

.72677 

.  1468 

37 

.64812 

•76154 

.66131 

.75011 

.67430 

•73846 

.68709 

•72657 

.69966 

•  '447 

36 

25 

.64834 

•76135 

.66153 

.  74992 

•67452 

.73826 

.68730 

•72637 

.69987 

•  '427 

35 

26 

.64856 

.76116 

.66175 

•74973 

•67473 

.73806 

.68751 

.72617 

.70008 

•  I4°7 

34 

27 

.64878 

.76097 

.66197 

•  74953 

•67495 

•73787 

.68772 

•72597 

.70029 

•  1386 

33 

28 

.64901 

.76078 

.66218 

•  74934 

•67516 

•73767 

.68793 

•72577 

.70049 

•  1366 

32 

29 

.64923 

.76059 

.66240 

•749J5 

•67538 

•73747 

.68814 

•72557 

.70070 

•  1345 

31 

.64945 

.7604! 

.66262 

.74896 

•67559 

•73728 

.68835 

•72537 

.70091 

•  1325 

30 

31 

.64967 

.76022 

.66284 

•74876 

.67580 

•73708 

.68857 

.72517 

.70  12 

•  '305 

29 

32 

.64989 

.76003 

.66306 

•74857 

.67602 

.73688 

•72497 

.70  32 

.  1284 

28 

33 

.65011 

•75984 

.66327 

.74838 

.67623 

•73669 

.  68899 

•72477 

•70  53 

.  1264 

27 

34 

•65033 

•75965 

-66349 

.74818 

.67645 

•73649 

.68920 

•72457 

.70  74 

•  1243 

26 

35 

.65055 

•  75946 

•66371 

•  74799 

.67666 

•73629 

.68941 

•72437 

•70  95 

•  1223 

25 

'v> 

•65077 

•75927 

.66393 

.74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.  1203 

24 

37 

.65100 

.75908 

.66414 

.74760 

.67709 

•73590 

.68983 

•72397 

.70236 

.  1182 

23 

38 

.65122 

•75889 

.66436 

•74741 

.67730 

•73570 

.69004 

•72377 

•70257 

.  1162 

22 

39 

.65144 

•75870 

.66458 

•74722 

.67752 

•7355 

.69025 

•72357 

.70277 

.  1141 

21 

4o 

.65166 

•75851 

.66480 

•74703 

•67773 

•7353 

.69046 

•72337 

.70298 

.  II2I 

20 

41 

.65188 

•75832 

.66501 

•74683 

•67795 

•7351 

.69067 

•72317 

•70319 

.71100 

9 

42 

.65210 

•75813 

.66523 

.74664 

.67816 

•7349 

.69088 

.72297 

•  70339 

.71080 

8 

43 

.65232 

•75794 

.66545 

•  74644 

•67837 

•7347 

.69109 

•72277 

.70360 

•71059 

I 

44 

•65254 

•75775 

.66566 

•74625 

.67859 

•73452 

.69130 

•72257 

.70381 

•71039 

6 

45 

.65276 

•75756 

.66588 

.74606 

.67880 

•73432 

.69151 

•72236 

.  70401 

.71019 

S 

*6 

•65298 

•75738 

.66610 

.74586 

.67901 

•734'3 

.69172 

.72216 

.70422 

.70998 

4 

47 

.65320 

•757J9 

.66632 

•74567 

•67923 

•73393 

.69193 

.72106 

•70443 

.70978 

3 

48 
49 

•65342 
.65364 

•75700 
.75680 

.66653 
.66675 

•74548 
.74528 

.67944 
•67965 

•73373 
•73353 

.69214 
•69235 

.72176 
.72156 

.  70463 
.70484 

•70957 
•70937 

2 

5° 

.65386 

.75661 

.66697 

•74509 

.67987 

•73333 

.69256 

•72136 

•70505 

.70916 

° 

j, 

•65408 

.75642 

.66718 

•74489 

.68008 

•733H 

.69277 

.72116 

•70525 

.70896 

8 

52 

.65430 

•75623 

.66740 

.74470 

.68029 

•73294 

.69298 

.72095 

.70875 

8 

53 

•65452 

•75604 

.66762 

•74451 

.68051 

•73274 

•69319 

•72075 

.70567 

•70855 

7 

54 

•65474 

•75585 

.66783 

•74431 

.68072 

•73254 

.69340 

•  72055 

.70587 

•70834 

6 

55 

•65496 

•75566 

.66805 

.74412 

.68093 

•73234 

.69361 

•72035 

.70608 

.70813 

5 

.65518 

•75547 

.66827 

•74392 

.68115 

•73215 

.69382 

.72015 

.70628 

•70793 

4 

i 

.65540 
65562 

•75528 

.66848 
.66870 

•74373 

.68136 

•73195 

-69403 

•71995 
.71974 

.70649 
.  70670 

•70772 
.  70752 

3 

w 

.65584 

•75490 

.66891 

•74334 

.68179 

•73*55 

•71954 

.70690 

•70731 

i 

.65606 

•7547' 

.  66g  i  3 

•74314 

.68200 

•73135 

.69466 

•71934 

.70711 

.  7071  1 

o 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

4< 

>° 

4* 

J° 

4 

1° 

4 

3° 

4. 

NATURAL  TANGENTS  AND  COTANGENTS. 


C 

> 

] 

2 

° 

1       : 

>° 

^ 

* 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

o 

.00000 

Infin. 

.01746 

57-2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

I 

.00029 

3437-75 

•01775 

56.35°6 

.03521 

28.3994 

.05270 

iS-9755 

.07022 

14.2411 

2 

.00058 

1718.87 

.01804 

55.4415 

•03550 

28.  1664 

.05299 

18.87,1 

.07051 

14.1821 

3 

.00087 

1145.92 

•01833 

54-5613 

•03579 

27.9372 

.05328 

,8.7678 

.07080 

14-1235 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

•05357 

18.6656 

.07110 

14.0655 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

•05387 

18.5645 

•07139 

14.0079 

6 

.00175 

572-957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

.07168 

-13-9507 

7 

491.106 

.01949 

51.3032 

.03696 

27.0566 

•°5445 

18-3655 

.07107 

13.8940 

8 
9 

.00262 

429.718 
381.971 

.01978 
.02007 

50.5485 
49.8157 

•03725 
•03754 

26.8450 
26.6367 

•05474 
•05503 

18.2677 
18.1708 

.07227 
•07256 

13-8378 
13.782, 

10 

.00291 

343-774 

.02036 

49.1039 

•03783 

26.4316 

•05533 

18.0750 

.07285 

,3.7267 

ii 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17  9802 

•07314 

13.6719 

12 

.00349 

286.478 

.02095 

47-7395 

.03842 

26.0307 

•05591 

17-8863 

•07344 

13.6174 

13 

.00378 

64.441 

.02124 

47.0853 

•03871 

25.8348 

.05620 

17.7934 

•07373 

13-5634 

H 

.00407 

45-552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

.07402 

13.5098 

11 

.00436 
.00465 

29.182 
14.858 

.02211 

45-8294 
45.2261 

.03929 
.03958 

25-4517 
25.2644 

.05678 
.05708 

17.6106 

•07431 
.07461 

13-4566 
13-4039 

17 

.00495 

.02240 

44.6386 

25.0798 

•°5737 

17-4314 

.07490 

13-3515 

18 

.00524 

90.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17-3432 

•07519 

,3.2996 

ig 

•00553 

80.932 

.02298 

43-5o8i 

.04046 

24.7185 

•05795 

17-2558 

.07548 

13.2480 

2O 

.00582 

71-885 

.02328 

42.9641 

•04075 

24-5418 

.05824 

17-1693 

•07578 

13.1969 

21 

.00611 

63.700 

•02357 

42-4335 

.04104 

24-3675 

•05854 

17.0837 

.07607 

13-1461 

22 

.00640 

56.259 

.02386 

41.9158 

•04133 

24-1957 

.05883 

16.9990 

.07636 

13.0958 

23 

.00669 

49.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

•07665 

13.0458 

24 

.00698 

43-237 

40.9174 

.04191 

23-8593 

.05941 

,6.8319 

.07695 

12.9962 

25 

.00727 

37-507 

•02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

•07724 

12.9469 

26 

.00756 

32.219 

.02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

•°7753 

,2.8981 

27 

.00785 

27.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

,6.5874 

.07782 

12.8496 

•8 

.00815 

22-774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

.07812 

12.8014 

29 

.00844 

18.540 

.02589 

38.6177 

•04337 

23-0577 

.06087 

,6:4283 

.07841 

12.7536 

3° 

.00873 

14.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

.07870 

12.7062 

31 

,00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

,6.2722 

.07899 

12.659, 

32 

.00931 

107.426 

.02677 

37-3579 

.04424 

.06175 

16.1952 

.07929 

,2.6,24 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.  06204 

16.  1190 

.07958 

12.5660 

34 

.00989 

101.107 

•02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

.07987 

12.5199 

35 

98.2179 

.02764 

36.1776 

.04512 

22.  1640 

.06262 

15-9687 

.08017 

12.4742 

36 

.01047 

.02793 

35.8006 

.04541 

22    0217 

.06291 

15.8945 

.08046 

,2.4288 

37 

.01076 

92-9085 

35-43I3 

.04570 

21.8813 

.  0632  i 

,5.  821! 

-08075 

,2.3838 

38 

.01105 

.02851 

35-0695 

.04599 

21.7426 

.06350 

T5.7483 

.08104 

12.339° 

39 

.01135 

88^436 

.02881 

34.7I5I 

.04628 

21.6056 

.06379 

15.6762 

.08134 

12.2946 

.01164 

85.9398 

.O291O 

34-3678 

.04658 

21.4704 

.06408 

15.6048 

.08163 

12.2505 

4' 
42 

.01193 

83.8435 

81.8470 

.02939 
.02968 

34-0273 
33.6935 

.04687 
.04716 

21.3369 
21.2049 

•06437 
.06467 

15.5340 
15-4638 

.08192 

.08221 

12.2067 
,2.1632 

43 

.01251 

79-9434 

.02997 

33-3662 

•04745 

21.0747 

.06496 

15-3943 

.08251 

44 

.01280 

78.1263 

.03026 

33-0452 

•04774 

20.9460 

.06525 

15-3254 

.08280 

12.0772 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

.06554 

15-2571 

.08309 

12.0346 

46 

.01338 

74-7292 

.03084 

32-4213 

.04833 

20:6932 

.06584 

15-1893 

•08339 

11.9923 

47 

.01367 

73-J390 

.03114 

.04862 

20.5691 

.06613 

15.1222 

.08368 

11.9504 

48 

.01396 

71-6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

•08397 

11.9087 
ii  .8673 

49 

5" 

•01455 

68.7501 

.03201 

31-2416 

.04949 

20.2056 

.06700 

14.9244 

:  08456 

i,  .'8262 

Si 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14-8596 

'08485 

".7853 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14-7954 

.08514 

"•7448 

53 

.01542 

64.8580 

.03288 

30.4116 

•05037 

I9-8546 

.06788 

14-7317 

.08544 

11.7045 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

.08573 

11.6645 

.01600 

.01629 

62.4992 
61.3829 

.03346 
.03376 

29.8823 
29.6245 

•05095 
.05124 

19.6273 

19.5156 

.'06876 

14.6059 
14.5438 

.08602 
.08632 

11.6248 
".5853 

57 

.01658 

60.3058 

.03405 

29-37I1 

•05153 

.06905 

14.4823 

.08661 

11.5461 

58 
59 

'.olfil 

59.2659 
58.2612 

•03434 
.03463 

28:8771 

.05182 

19.2959 
19.1879 

.069-54 
.06963 

14.4212 
14.3607 

.08690 
.08720 

11.5072 
11.4685 

fa 

.01746 

57.2900 

.03402 

28.6363 

.05241 

19.0811 

.06993 

14-3007 

.08749 

11.4301 

, 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

8( 

1° 

8* 

;° 

8; 

8( 

)° 

8^ 

0 

) 

NATURAL  TANGENTS  AND  COTANGENTS. 


33 


5 

0 

6 

0 

7° 

s 

9 

0 

71 

' 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

I 

* 

.08749 
.08778 

,1.4301 
11.3919 

-  05,0 
-  0540 

9.5,436 
9.4878, 

.12278 
.12308 

8.14435 
8.1248, 

:  4*084 

7-"537 
7.10038 

.15838 
.  ,  5868 

6-31375 
6.30189 

60 

59 

2 

.08807 

11.3540 

.  0569 

0.4614, 

.12338 

8.10536 

•   4"3 

7.08546 

.15898 

6.29007 

53 

3 

.08837 

,1.3163 

•  0599 

9-435'S 

.12367 

8.08600 

•   4M3 

7.07059 

-15928 

6.27829 

57 

4 

.08866 

,1.2789 

.  0628 

9.40904 

.12397 

8.06674 

4'73 

7-05579 

6.26655 

56 

5 

.08895 

11.2417 

-  0657 

9.38307 

.12426 

8.04756 

.     42C2 

7.04105 

.  I  5988 

6.25486 

55 

6 

.08925 

11.2048 

.  0687 

9-35724 

.12456 

•     4232 

7.02637 

.l6ol7 

6.24321 

54 

7 

ii  .1681 

.  07,6 

9.33I55 

.12485 

8.00948 

.     4262 

7-01174 

.16047 

6.23160 

53 

8 

08983 

ii.  ,3.6 

.  0746 

9-30599 

•'2515 

.99058 

•     4291 

6.99718 

.16077 

6.22003 

5-! 

9 

090,3 

11.0954 

•  0775 

9.28058 

•12544 

.97,76 

•     4321 

6.98268 

.T6lO7 

6.20851 

5' 

09042 

11.0594 

.  0805 

9-25530 

•12574 
,060:1 

•95302 
•93438 

•     4351 

6.96823 
6.95385 

.16137 

6.19703 

50 

'3 

09,01 
.09130 

0.9882 
0.9529 

.  0863 
.  0893 

9.20516 

.12003 
.12633 
.,2662 

.91582 
•89734 

•     4440 

6-93952 
6.92525 

!l6226 

6.18559 
6.174,9 
6.16283 

40 

48 

47 

M 

.09159 

0.9,78 

.  0922 

9-15554 

.,2692 

.87895 

•     447° 

6.91104 

.  16256 

6.15151 

46 

1  6 

.09,89 
.092,8 

0:8483 

•  0952 

.  0981 

9.13093 
9.10646 

.12751 

.86064 
.84242 

•    4499 
•    4529 

6  .  89688 
6.88278 

.16286 
.16316 

6.14023 
6.12899 

45 
44 

'7 

•09247 

9.08211 

.,2781 

.82428 

•   4559 

6.86874 

.16346 

6.11779 

43 

18 

.09277 

0.7797 

.     040 

9.05789 

.80622 

.    4588 

6.85475 

.'6376 

6.10664 

42 

10 

.09306 

0-7457 

.     070 

9-03379 

:  12840 

.78825 

.   4618 

6.84082 

.16405 

6.09552 

41 

20 

.09335 

0.71,9 

•     099 

9.00983 

.12869 

•77°35 

.   4648 

6.82694 

.16435 

6.08444 

40 

21 

.09365 

0.6783 

,28 

8.98598 

.12899 

•75254 

.   4678 

6.81812 

.16465 

6.07340 

39 

22 

.09394 

0.6450 

-     158 

8.96227 

.12929 

.73480 

•   4707 

6.79936 

•  '6495 

6.06240 

38 

2  } 

.09423 

.     187 

8.93867 

.12958 

•7I7'5 

•   4737 

6.78564 

.16525 

6.05,43 

37 

24 

•09453 

o:5789 

.     217 

8.91520 

.12988 

.69957 

•   4767 

6.7719° 

-'6555 

6.04051 

36 

25 

.09482 

0.5462 

.     246 

8.89,85 

.,30,7 

.68208 

•   4796 

6  •  75838 

.16585 

6.02962 

35 

20 

.09511 

0.5136 

.     276 

8.86862 

.,3047 

.66466 

.   4826 

6.74483 

.16615 

6.01878 

34 

27 

.0954, 

0.48,3 

-     3°5 

8.8455, 

.13076 

•64732 

.   4856 

6-73133 

.16645 

6.00797 

33 

28 

.09570 

0.4491 

•     335 

8.82252 

.,3,06 

.63005 

.   4886 

6.71789 

.16674 

5.99720 

3* 

29 

.09600 

0.4172 

•     364 

8  .  79964 

.13,36 

.6,287 

•   49'5 

6.70450 

.  16704 

5-98646 

31 

3° 

.09629 

0.3854 

•     394 

8.77689 

.13,65 

•59575 

•   4945 

6.69116 

•'6734 

5-97576 

30 

31 

.09658 

0.3538 

•      423 

8.75425 

•13195 

•57872 

•   4975 

6.67787 

.16764 

5.96510 

29 

32 

.09688 

0.3224 

•     452 

8  73172 

•13224 

.56176 

•   5005 

6.66463 

.16794 

5-95448 

33 

.097,7 

0.29,3 

.     482 

8.7093, 

•'3254 

•54487 

•   5034 

6.65144 

.16824 

5.94390 

34 

.09746 

0.2602 

8.68701 

.,3284 

.52806 

.   5064 

6.6383, 

.16854 

5-93335 

35 

.09776 

0.2294 

•      54' 

8.66482 

.133,3 

.51132 

•   5094 

6.62523 

.16884 

5.92283 

36 

.09805 

0.1988 

.      570 

8.64275 

•'3343 

•49465 

6.61219 

.16914 

5-91236 

37 

.09834 

0.1683 

.     600 

8.62078 

.13372 

.47806 

•   5153 

6.59921 

.16944 

5.90191 

38 

.09864 

0.1381 

629 

8.59893 

•13402 

.46154 

•   5183 

6.58627 

.16974 

5-89151 

39 

.09893 

0.1080 

•     659 

8.57718 

•13432 

•44509 

•   5213 

6-57339 

.17004 

5.88114 

.09923 

0.0780 

.     688 

8-55555 

•13461 

.42871 

•   5243 

6.56055 

•17033 

5.87080 

4' 

.09952 

10.0483 

.     718 

8.53402 

•I349I 

.41240 

•    5272 

6-54777 

.17063 

5.86051 

42 

.0998! 

•     747 

8.51259 

.13521 

.39616 

•   5302 

6-53503 

•17093 

5.85024 

43 

9.9893, 

•     777 

8.49,28 

•'355° 

•37999 

6.52234 

.17123 

5.84001 

44 

.  0040 

9.96007 

.     806 

8.47007 

•13580 

•36389 

.   5362 

6.50970 

5.82982 

45 

.  0069 

9.93,01 

•     836 

8.44896 

.13609 

.34786 

•   5391 

.17183 

5.81966 

46 

.  0009 

9.90211 

.     865 

8.42795 

•'3639 

•33190 

•   5421 

6.48456 

.17213 

5-80953 

47 

.  0128 

9-87338 

•     895 

8.40705 

.13669 

.31600 

•   545' 

6.47206 

•17243 

5-79944 

48 

0158 

9.84482 

8.38625 

.13698 

.30018 

-   548' 

6.45961 

.17273 

5  •  78938 

49 

.  0187 

9.8,64, 

•      954 

8.36555 

.13728 

.28442 

•  55" 

6.44720 

•17303 

5-77936 

5" 

.10216 

9.78817 

-      983 

8.34496 

•'3758 

.26873 

•   5540 

6.43484 

•'7333 

5-76937 

51 

.10246 

9.76009 

-      013 

8  .  32446 

.13787 

.25310 

-  557° 

6.42253 

.17363 

5-7594' 

52 

•  '0275 

9.73217 

.      042 

8.30406 

.138,7 

•23754 

.  5600 

6.41026 

•'7393 

5  .  74949 

53 

.  10305 

9  .  7044  1 

.      072 

8.28376 

.13846 

.  22204 

•   5630 

6.39804 

.17423 

5-7396o 

54 

55 

•  10334 
•  0363 

9.67680 
9-64935 

8-26355 
8-24345 

.13876 
.13906 

.20661 
.19125 

.   5660 
.  5689 

6.38587 
6-37374 

•'7453 
•17483 

5  •  72974 
5.71992 

5« 

•  I0393 

9.62205 

:  160 

8.22344 

•'3935 

•'7594 

•  57'9 

6.36165 

•I75I3 

5-7IOI3 

57 

.  10422 

9-5949° 

.  190 

8.20352 

.'3965 

.16071 

•   5749 

6.34961 

•'7543 

5  .  70037 

5« 

.   0452 

9.56791 

.  219 

8.18370 

•'3995 

•'4553 

•   5779 

6.33761 

•'7573 

5.69064 

59 

.10481 

9-54'o6 

•  249 

8.16398 

.14024 

.  13042 

.   5809 

6.32566 

.17603 

5.68094 

Go 

.105,0 

9.5M36 

.  278 

8.14435 

.14054 

7."537 

•   5838 

6.31375 

•17633 

5.67128 

, 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

8 

4° 

8 

3° 

8 

2° 

8 

t° 

8( 

D° 

34 


NATURAL  TANGENTS  AND  COTANGENTS. 


I 

0° 

I 

1° 

i 

2° 

I 

3° 

i 

4° 

71 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

*  i 

° 

:!7766 

5-67128 
5-66165 

.19438 
.19468 

5-M455 
5-13658 

.21256 

.2    286 

4.70463 
4.69791 

.23087 
.23117 

•33148 
•32573 

•24933 
.24964 

.00582 

777 

59 

2 

.1769 

5.65205 

-19498 

5.12862 

.2    3,6 

4.69121 

.23148 

.32001 

•24995 

.00086 

58 

3 

.1772 

5.64248 

•19529 

5.12069 

•2  347 

4-68452 

•23179 

•31430 

.25026 

.90592 

57 

4 

•  1775 

5-63295 

5.11279 

•2   377 

4.67786 

.23209 

.30860 

.25056 

.99099 

56 

5 

.1778 

5.62344 

•19589 

5  •  10490 

.2    408 

4.67121 

.23240 

.30291 

.25087 

.98607 

55 

6 

.17813 

5.61307 

.19619 

5.09704 

•2    438 

4.66458 

.23271 

•29724 

.25118 

.981,7 

54 

7 

•17843 

5.60452 

.19649 

5.08921 

.2    469 

4.65797 

.23301 

•29159 

.25,49 

.97627 

53 

8 

.17873 

5-595" 

.  19680 

5.08139 

.2  499 

4.65138 

•23332 

•28595 

.25,80 

.97,39 

52 

9 

•17903 

.197,0 

5-07360 

•2    529 

4.64480 

•23363 

.28032 

.252,1 

-96651 

51 

JO 

•17933 

5-  '57638 

.19740 

5-06584 

.2560 

4-63825 

•23393 

•27471 

.25242 

3.96165 

50 

12 

.17963 

.17993 
.18023 

5.56706 

5-55777 
5.54851 

.19770 
.19801 
.19831 

5.05809 
5-05037 
5.04267 

.2    590 
.2    621 
.2    65, 

4.6317, 
4.625,8 
4.6,868 

•23424 
•23455 
•23485 

.26911 
•26352 
•25795 

•25273 
.25304 
•25335 

3-9568o 
3.95,96 
3-94713 

9\ 

47 

,4 

.18053 

5-53927 

.19861 

5-03499 

.2    682 

4.6,2,9 

.23516 

•25239 

•25366 

3-94232 

46 

16 

.18083 
18113 

5-53007 

.1989, 

5-02734 

.2     712 

4.60572 

•23547 

•24685 

•25397 

3-9375' 

45 

17 

-18-43 

5-'  5  1*176 

•19952 

5.01210 

•2   773 

4-59283 

•  2364.8 

•23580 

•25459 

3-92793 

43 

18 

.18173 

5-50264 

.  19982 

5.00451 

.2    804 

4.5864, 

•23639 

.23030 

•2549° 

3.92316 

42 

20 

.18203 
.18233 

5.49356 
5-48451 

.20042 

4.99695 
4.98940 

.2    834 
.2864 

4.5800, 
4-57363 

.23670 
•23700 

.2248, 
•21933 

•25521 
•25552 

3-91839 
3.9,364 

4' 

40 

2, 

22 

.18263 
.18293 

5.47548 
5.46648 

.20073 

4.98188 
4.97438 

.2,895 
•21925 

4.56726 
4.5609, 

•23731 
•23762 

.21387 

.20842 

•25583 
.256,4 

3.90890 

3.90417 

P 

23 

.18323 

5-45751 

•20133 

4.96690 

.21956 

4-55458 

•23793 

.  20298 

.25645 

3-89945 

37 

24 

•'8353 

.20164 

4-95945 

.21986 

4-54826 

.23823 

•19756 

.25676 

3.89474 

30 

25 

.18384 

5  .  43966 

.20194 

4.95201 

.22017 

4.54,96 

•23854 

•19215 

•25707 

3.89004 

35 

26 

.18414 

5-43077 

4.94460 

.22047 

4-53568 

•23885 

.18675 

•25738 

3-88536 

34 

27 

.18444 

5.42192 

.70254 

4-93721 

.22078 

4.5294, 

•23916 

.18137 

.25769 

3.88068 

33 

28 

.18474 

5.41309 

.2^285 

4.92984 

.22108 

4.52316 

•23946 

.17600 

.25800 

3.8760, 

32 

29 

3° 

.  18504 
.18534 

5.40429 
5-39552 

•20315 
•20345 

4.92249 
4.91516 

-22139 
.22169 

4.51693 
4.51071 

•23977 
.24008 

.  .17064 
•  16530 

.25831 

.25862 

3.87136 
3.86671 

31 

30 

32 
33 

.18564 
•18594 
.18624 

5-38677 
5-37805 
5-36936 

•20376 
.20406 
.20436 

4.90785 
4.90056 
4.89330 

.2226 

4.50451 
4.49832 
4-49215 

•  24039 
.24069 
.24100 

•  15997 
•15465 
•14934 

•25893 
•25924 
•25955 

3-85745 
3-85284 

3 

27 

34 

.18654 

5.36070 

.20466 

4.88605 

.2229 

4.48600 

•24,31 

•  14405 

.  25986 

{.84824 

26 

35 

.18684 

5.35206 

.20497 

4.87882 

.2232 

4-47986 

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3-84364 

25 

36 

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5-34345 

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4.87162 

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4-47374 

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3.83906 

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37 

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5.33487 

4.86444 

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4  .  46764 

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3.83449 

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38 

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5-32631 

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4-85727 

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4.46155 

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3.82992 

22 

39 

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5.3I778 

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4.85013 

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4-45548 

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3-82537 

21 

40 

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5-30928 

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4.84300 

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4.44942 

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3-82083 

20 

41 

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5.30080 

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4-8359° 

.22505 

4.44338 

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•  10736 

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3-81630 

19 

42 

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5-29235 

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4.82882 

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4-43735 

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3-81,77 

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43 

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5-28393 

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4.82175 

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4-43'34 

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3.80726 

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44 

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5-27553 

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4.81471 

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4.42534 

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3.80276 

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5-26715 

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4.80769 

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4.41936 

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3.71,827 

15 

46 

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5-25880 

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4.80068 

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4.41340 

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3-79378 

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47 

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5.25048 

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4-79370 

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4.40745 

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3-78931 

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5.24218 

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4.78673 

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4.40152 

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3.78485 

12 

49 

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5-23391 

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4.77978 

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4.3956o 

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3.78040 

II 

50 

.19136 

5-22566 

.20952 

4.77286 

.22781 

4.38969 

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3-77595 

1° 

5' 

.19166 

5.21744 

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4-76595 

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4-38381 

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3-77152 

9 

52 

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5.20925 

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4.75906 

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4-37793 

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3.76709 

8 

53 

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5.20107 

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4.75219 

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4.37207 

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3.76268 

7 

54 

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5-I9293 

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4-74534 

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4-36623 

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3-75828 

6 

.19287 

5.18480 

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4-73851 

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4.36040 

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3-75388 

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5-17671 

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4-73170 

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4-35459 

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3-7495° 

57 

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5.16863 

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4.72490 

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4  .  34879 

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3-74512 

58 

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5.16058 

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4.71813 

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4-34300 

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3  •  74075 

59 

.19408 

5-15256 

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4-71137 

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4-33723 

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.01576 

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3  •  73640 

60 

.19438 

5-14455 

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4.70463 

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4-33I48 

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3-73205 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

, 

7 

f 

7 

8° 

7 

7° 

7 

6° 

7 

5° 

NATURAL  TANGENTS  AND  COTANGENTS. 


I 

5° 

I 

I 

7° 

I 

8° 

I 

; 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

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3-73205 

.28675 

3.4874. 

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3-27085 

•  32492 

3.07768 

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1 

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3.72771 

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3-48359 

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3-26745 

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3.07464 

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59 

2 

.26857 

3-72338 

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3-47977 

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3  .  26406 

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3.07.60 

•34498 

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3 

.26888 

3.7.907 

.  8769 

3-47596 

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3.26067 

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3.06857 

•34530 

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mm 

4 

.26920 

3-71476 

.  8800 

3.47216 

.30700 

3.25729 

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3-06554 

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mA 

5 

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3.7.046 

•  8832 

3-46837 

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3.06252 

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55 

6 

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3.70616 

.  8864 

3-46458; 

•30764 

3-25055 

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3.05950 

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54 

7 

g 

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3.70188 
3.69761 

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3.46080 

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3-247-9 

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3-05649 

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.  8824O 

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9 

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3-69335 

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3.45327 

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3.24049 

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3*05349 
3.05049 

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51 

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3.68909 

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3-  4495  - 

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3-237-4 

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3-04749 

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3-68485 

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3-44576 

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3-2338- 

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3-0445° 

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M 

12 

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3.6806. 

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3.44202 

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3.23048 

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3.04152 

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VJ 

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3.67638 

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3.43829 

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3.227.5 

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3.03854 

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3-67217 
3.66795 

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3-43456 
3.43084 

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3-22384 
3.22053 

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3.03260 

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1  6 

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3.66375 

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3-42713 

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3.21722 

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3.02963 

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44 

17 

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3-65957 

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3.42343 

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3.2.392 

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3.02667 

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3-65538 

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3.21063 

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3-02372 

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19 

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3-65-21 

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3.41604 

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3.20734 

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3.02077 

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3-64705 

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3.41236 

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3.20406 

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3.0.783 

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3.64289 

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3-40869 

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3.20079 

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3-0.489 

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3-63874 

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3.63048 
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3-39771 
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3.19100 
3.I8775 

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3.006,1 
3.00319 

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3.6.8.4 
3.6.405 

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3.38679 
3-383I7 

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3.18,27 
3-17804 

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2.99738 
2-99447 

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83,76 

33 

52 

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3.60996 

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3-37955 

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3.1748. 

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2.99.58 

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2.98868 

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3-37234 

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3.16838 

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32 

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3-59775 

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3-36875 

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3.165-7 

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8.870 

20 

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3-5937° 

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3.36516 

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3.16.97 

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8.6.0 

27 

34 

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3-36158 

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3-I5877 

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8-350 

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3.58.60 

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3-35443 

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3.15240 

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3.57758 

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3-35087 

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3.14922 

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3-57357 

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3-34732 

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3.14605 

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39 

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3-56957 
3-56557 

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3-34377 
3-34023 

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3.14288 
3.I3972 

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3.56.59 

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3-33670 

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3.13656 

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79545 

10 

42 

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3  .55761 

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3-333I7 

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3-1334- 

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3-55364 

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3.32965 

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3.13027 

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3.54968 

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3-326.4 

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3-  -27-3 

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78778 

1  6 

45 

3-54573 

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3.32264 

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3.12400 

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3-3I9I4 

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3.12087 

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14 

47 

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3-53393 

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3-3I565 
3-312.6 

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3-1-775 
3.1.464 

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78014 
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13 

12 

40 

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3-53001 

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3-30868 

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3.  1-153 

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II 

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3.52609 

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3.30521 

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3.10842 

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51 

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3.522.9 
3.51829 

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3-30174 

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3.10532 

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2.77002 
2.76750 

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3-29483 

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3.099.4 

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2.76498 

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54 

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3-51053 

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3.29.39 

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3,09606 

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2.76247 

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3.50666 

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3.09298 

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2-75996 

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3.50279 
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2.75746 
2.75496 

4 

3 

58 

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3.49509 

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3-27767 

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3-o8379 

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2.75246 

2 

59 

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3-49I23 

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3.27426 

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3.08073 

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2.74097 

I 

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.28675 

3.48741 

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3.27085 

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3.07768 

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2.74748 

(1 

, 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

, 

7 

*° 

7: 

)° 

7- 

J° 

7 

[° 

7C 

NATURAL  TANGENTS  AND  COTANGENTS. 


2 

o° 

2 

i° 

2 

2° 

2 

3° 

2 

4° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

•36397 

2.  4748 

•38386 

2.60509 

.40403 

2.47509 

.42447 

2-35585 

•44523 

.24604 

60 

I 

.36430 

2.   4499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2-35395 

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.24428 

59 

2 

.36463 

2.    4251 

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2.60057 

.40470 

2.47095 

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2.35205 

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.24252 

58 

3 

•36496 

2.    4004 

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2.59831 

.40504 

2.46888 

•42551 

2-35015 

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•24077 

57 

4 

.36529 
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2.    3756 
2.     3509 

•38520 
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2.59606 
2.59381 

.40538 

2.46682 
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2.34825 
2-34636 

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•  3902 
.  3727 

56 
55 

6 

•  36595 

2.    3263 

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2.59156 

.40606 

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2-34447 

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•   3553 

54 

7 

.36628 

2.     3017 

.38620 

2.58932 

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2.34258 

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•    3378 

53 

8 

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2.    2771 

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2.58708 

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2  .  34069 

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•    3204 

52 

9 

.36694 

2.     2526 

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2.58484 

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•45655 

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2.33881 

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-   3030 

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2.     228l 

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2.58261 

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•45451 

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2.33693 

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50 

ii 

.36760 

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2.    2036 

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58787 

2:58038 

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.45246 

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2.33505 

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49 

M 

.36826 
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2.     1792 
2.     I548 
2.     1305 

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2-57593 
2-57371 

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2.33130 
2.32943 

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a 

15 

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2.     1062 

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2-57150 

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2.32756 

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45 

16 

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2.70819 

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2.56928 

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2.32570 

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44 

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2.70577 

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2.56707 

2  .  56487 

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2.32383 

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43 

tg 

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2.7O335 
2.70094 

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2.56266 

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2.32012 

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M 

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2.69853 
2    60612 

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2.56046 

2.55827 

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2.31826 

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40 

22 
23 

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2.09012 
2.69371 
2.69131 

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2.55608 
2.55389 

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2.31456 
2.31271 

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38 
37 

24 

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2.68892 

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2.55170 

•  4  217 

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2.31086 

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36 

25 

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2.68653 

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2.54952 

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2.30902 

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35 

26 

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2.68414 

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2-54734 

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2.30718 

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34 

27 

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2.68175 

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2.54516 

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2.30534 

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28 

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2.67937 

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2.54299 

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2-30351 

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32 

29 

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2.67700 

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2.54082 

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2.30167 

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2.67462 

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2.53865 

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2.29984 

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30 

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2.67225 

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2.53648 

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2.41223 

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2.66989 

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2.53432 

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2.41025 

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2.29619 

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28 

33 

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2.66752 

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2.53217 

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2.40827 

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2.29437 

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27 

34 

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2.66516 

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2  .  53001 

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2.40629 

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2.20254 

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2.66281 

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2.52786 

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2.40432 

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2.29073 

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25 

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2.66046 

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2.5257I 

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2.40235 

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2.28891 

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24 

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2.65811 

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2.52357 

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2.40038 

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2.28710 

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2.28528 

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2.65342 

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2.51929 

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2.28348 

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2.39253 

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2.27987 

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2.51289 

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2.27806 

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18 

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2.64410 

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2.51076 

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2.38863 

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2.27626 

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17 

44 

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2.64177 

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2.50864 

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2.38668 

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NATURAL  TANGENTS  AND  COTANGENTS. 


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.66428 

.62446 
.62487 

.60137 
1.60033 

.64899 
.64941 

•54085 
.53986 

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•67451 

.48349 
.48256 

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.7002. 

.42903 
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0 

, 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

, 

5< 

?° 

5 

3° 

s 

7° 

5 

5° 

5. 

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) 

NATURAL  TANGENTS  AND  COTANGENTS. 


39 


3 

5° 

3 

5° 

3 

7° 

3 

3° 

3 

9° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

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.42726 

•72654 
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.  32624 

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•  27994 
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to 

59 

.70.07 

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1-37470 

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.2784. 

.81075 

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58 

.70151 

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57 

•  70194 

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23.96 

56 

5 

.70238 

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55 

c 

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54 

7 

.70325 

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53 

8 

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52 

9 

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5' 

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5° 

H 

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13 

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.8,606 

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47 

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46 

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17 

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.8.946 

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21 

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39 

22 

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38 

23 

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37 

24 

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.21742 

36 

25 

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35 

26 

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34 

27 

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33 

28 

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29 

.7.285 

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31 

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20 

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28 

33 

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27 

34 

•  39850 

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35 

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9 

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51 

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4 

57 

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0 

t 

Cotang 

Tang 

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Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

1 

5' 

t° 

5 

3° 

5 

2° 

5 

L° 

5 

D° 

40 


NATURAL,  TANGENTS  AND  COTANGENTS. 


4 

3° 

4 

i° 

4 

2° 

4 

3 

4 

4° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

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56 

5 

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6 
7 

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54 
53 

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52 

9 

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5' 

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f 

' 

4 

9° 

4 

8° 

4 

7° 

4 

6° 

4 

5° 

TRAVERSE  TABLES 

OR 

LATITUDES ^ DEPARTURES  OF  COURSES 

CALCULATED  TO 

THREE  DECIMAL  PLACES 

FOR 

EACH   QUARTER  DEGREE   OF   BEARING 


LATITUDES   AND    DEPARTURES. 


43 


1 

I 

5 

1 

J 

5 

I 

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Lat. 

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1-375 

6.814 

1.604 

7.787 

1.834 

8.760 

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6-799 

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8.742 

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6.792 

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7.762 

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8-733 

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1.969 

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5.802 

1.528 

6.769 

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7-736 

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1-553 

6.761 

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7.727 

2.071 

8.693 

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1.603 

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7.709 

2.138 

8.673 

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5-775 

1.629 

6-737 

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7-700 

2.172 

8.662 

2-443 

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5-768 

1.654 

6.729 

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7.690 

2.205 

8.651 

2.481 

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1-399 

5.760 

1.679 

6.720 

1-959 

7.680 

2.239 

8.640 

2.518 

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1.420 

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1.704 

6.712 

1.988 

7.671 

2.272 

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1.729 

6.703 

2.017 

7.661 

2.306 

8.618 

2-594 

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1.462 

5-738 

1-754 

6.694 

2.047 

7.650 

2-339 

8.607 

2.631 

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1.483 

5-730 

1-779 

6.685 

2.076 

7.640 

2.372 

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1.504 

5-722 

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6.676 

2.105 

7-630 

2.406 

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1.524 

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1.829 

6.667 

2.134 

7.619 

2.439 

8.572 

2-744 

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1-545 

5.706 

1.854 

6.657 

2.163 

7.608 

2.472 

8.560 

2.781 

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1.566 

5-698 

1.879 

6.648 

2.192 

7.598 

2-505 

8-547 

2.818 

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1-587 

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1.904 

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2.221 

7.587 

2.538 

8-535 

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1.607 

5.682 

1.929 

6.629 

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7-575 

2.572 

8.522 

2.893 

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1.628 

5-673 

1-953 

6.619 

2.279 

7-564 

2.605 

8.510 

2.930 

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1.648 

5-665 

1.978 

6.609 

2.308 

7-553 

2.638 

8-497 

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5-656 

2.003 

6.598 

2-337 

7-541 

2.670 

8.484 

3-004 

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1.690 

5-647 

2.028 

6.588 

2.365 

7-529 

2.703 

8.471 

3.041 

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1.710 

5-638 

2.052 

6-578 

2-394 

7.518 

2.736 

8-457 

3.078 

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I-73I 

5-629 

2-077 

6.567 

2.423 

7.506 

2.769 

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5.620 

2.IOI 

6-557 

2.451 

7-493 

2.802 

8.430 

3-152 

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1.771 

5.611 

2.126 

6.546 

2.480 

7.481 

2.834 

8.416 

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2.150 

6-535 

2.509 

7.469 

2.867 

8.402 

3-225 

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5-592 

2-175 

6.524 

2-537 

7.456 

2.900 

8.388 

3.262 

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1-833 

5-582 

2.199 

6.513 

2.566 

7-443 

2.932 

8-374 

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1-853 

5-573 

2.223 

6.502 

2-594 

7-430 

2.964 

8-359 

3-335 

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1-873 

5-563 

2.248 

6.490 

2.622 

7-417 

2-997 

8-345 

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5-553 

2.272 

6-479 

2.651 

7-404 

3.029 

8-330 

3-408 

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7-364 

3.126 

8.285 

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1.974 

5-513 

2.368 

6.432 

2.763 

7-350 

3-158 

8.269 

3-553 

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1.994 

5-502 

2.392 

6.419 

2.791 

7-336 

3.190 

8.254 

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2.416 

6.407 

2.819 

7.322 

3.222 

8.238 

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2.034 

5.481 

2.440 

6-395 

2.847 

7.308 

3-254 

8.222 

3.661 

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2.054 

5-471 

2.464 

6.382 

2.875 

7-294 

3.286 

8.206 

3-696 

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2.073 

5-460 

2.488 

6.370 

2.903 

7.280 

3.318 

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5-449 

2.512 

6-357 

2.931 

7.265 

3-349 

8.173 

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2.113 

5-438 

2-536 

6-344 

2.958 

7.250 

3-381 

8.157 

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2-133 

5-427 

2-559 

6-331 

2.986 

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3.4I3 

8.140 

3-839 

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2.583 

6.318 

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2.172 

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2.607 

6-305 

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7.206 

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2.  192 

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2.630 

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0.895 

0.446 

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0.892 

2.685 

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0-454 

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0.466 

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0.931 

2.655 

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3-540 

1.862 

4-425 

62V 

28 

0.883 

0.469 

.766 

0-939 

2.649 

.408 

3-532 

1.878 

4-4I5 

62' 

28V 

0.881 

0-473 

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0-947 

2.643 

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3-524 

1.893 

4.404 

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0.879 

0-477 

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0-954 

2.636 

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3-5I5 

1.909 

4-394 

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0.877 

0.481 

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0.962 

2.630 

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3-507 

1.924 

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0.875 

0.485 

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0.970 

2.624 

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3-498 

1-939 

4-373 

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0.872 

0.489 

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0-977 

2.617 

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3-490 

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0.492 

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0.496 

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2.605 

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1.985 

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0.864 

0.504 

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2.592 

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2.015 

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3-429 

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2.565 

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3.420 

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4-275 

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0.853 

0.522 

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2.558 

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2.090 

4.263 

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0.848 

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0-530 

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3.401 
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2.105 

2.120 

4-252 
4.240 

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0.846 

0-534 

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2-537 

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2.134 

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0.839 

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2.516 

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3-355 

2-179 

4.193 

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2.509 

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3-345 

2.193 

4.181 

56V 

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0.834 

0-552 

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2.502 

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3-336 

2.208 

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0.831 

0.556 

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2-494 

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3-326 

2.222 

4-157 

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34 

0.829 

0-559 

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2-487 

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3-3i6 

2-237 

4.145 

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34V 

0.827 

0-563 

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2.480 

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3-306 

2.251 

4-133 

55V 

0.824 

0.566 

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2.472 

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3-297 

2.266 

4.121 

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0.822 

0.570 

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2.465 

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2.280 

4.108 

55V 

35 

0.819 

0-574 

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:  3-277 

2.294 

4.096 

55 

35V 

0.817 

0-577 

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2.450 

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3.267 

2.309 

4-083 

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0.814 

0.581 

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2.442 

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3-257 

2.323 

4.071 

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35  £ 

0.812 

0.584 

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2-435 

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3.246 

2-337 

4-058 

54V 

36 

0.809 

0.588 

.618 

.176 

2.427 

•763 

3-236 

2-351 

4-045 

54" 

36V 

0.806 

0.591 

-613 

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2.419 

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3.226 

2.365 

4.032 

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0.804 

0-595 

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2.412 

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3-215 

2-379 

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0.801 

0.598 

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2.404 

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2-396 

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0.796 

0.605 

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0-793 

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0.791 

0.612 

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2.372 

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3-163 

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0.788 

0.616 

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2-364 

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3-152 

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0.619 

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2.356 

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2.476 

3-927 

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0.783 

0.623 

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2.348 

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3.120 

2.504 

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3.109 

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LATITUDES  AND  DEPARTURES. 


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Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

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1 

26" 

2.192 

5-393 

2.630 

6.292 

3-069 

7.190 

3-507 

8.089 

3-945 

64° 

26^ 

2.  211 

5-38I 

2.654 

6.278 

3-096 

7-175 

3.538 

8.072 

3.981 

633/ 

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2.231 

5-370 

2.677 

6.265 

3-123 

7.IDO 

3-570 

8-054 

4.016 

63  K 

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5-358 

2.701 

6.251 

3-I5I 

7-144 

3.601 

8.037 

4.051 

63  % 

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5-346 

2.724 

6.237 

3.178 

7.128 

3.632 

8.019 

4.086 

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27^ 

2.289 

5-334 

2-747 

6.223 

3.205 

7.112 

3-663 

8.001 

4.121 

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2.309 

5.322 

2.770 

6.209 

3-232 

7.096 

7-983 

4.156 

62  yz 

27* 

2.328 

5.310 

2-794 

6-195 

3-259 

7.080 

3-725 

7-965 

4.190 

62  tf 

28" 

2-347 

5.298 

2.817 

6.181 

3.286 

7.064 

7-947 

4.225 

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28^ 

2-367 

5-285 

2.840 

6.166 

3-3I3 

7.047 

3.787 

7.928 

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2.386 

5-273 

2.863 

6.152 

3-340 

7.031 

3.817 

7.909 

4.294 

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2.405 

5.260 

2.886 

6-137 

3-367 

7.014 

3-848 

7.891 

4-329 

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29° 

2.424 

5-248 

2.909 

6.122 

3-394 

6-997 

3.878 

7.872 

61° 

29^ 

2-443 

5-235 

2.932 

6.107 

3.420 

6.980 

3.909 

7-852 

4-398 

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2.462 

5-222 

2-955 

6.093 

3-447 

6.963 

3-939 

7.833 

4-432 

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29  X 

2.481 

5-209 

2-977 

6.077 

3-474 

6.946 

3-970 

7.814 

4.466 

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30° 

2.500 

5.196 

3.000 

6.062 

3.500 

6.928 

4.000 

7-794 

4-500 

60 

30^ 

2.519 

5-I83 

3-023 

6.047 

3.526 

6.911 

4.030 

7-775 

4-534 

59^ 

30  y2 

2-538 

5-170 

3-045 

6.031 

3-553 

6.893 

4.060 

7-755 

4-568 

30% 

2.556 

5.I56 

3.068 

6.016 

3-579 

6.875 

4.090 

7-735 

4.602 

59^ 

31° 

2-575 

5-143 

3.090 

6.000 

3.605 

6.857 

4.120 

7-7I5 

4-635 

59° 

3I% 

2-594 

5.129 

5.984 

3-631 

6.839 

4.150 

7.694 

4-669 

5334- 

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2.612 

5-II6 

3-135 

5-968 

3-657 

6.821 

4.180 

7.674 

4.702 

58^ 

31  x 

2.631 

5-102 

3-157 

5-952 

3-683 

6.803 

4.210 

7-653 

4-736 

5Stf 

32 

2.650 

5.088 

3-180 

5.936 

3-709 

6.784 

4-239 

7.632 

4.769 

58 

32% 

2.668 

5-074 

3.202 

5-920 

3-735 

6.766 

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7.612 

4-802 

573^ 

32^ 

2.686 

5.060 

3-224 

5-904 

3.761 

6-747 

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7-591 

4-836 

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32  X 

2.705 

5-046 

3.246 

5.887 

3-787 

6.728 

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7-569 

4.869 

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2.723 

5.032 

3.268 

5-87I 

3-812 

6.709 

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7-548 

4.902 

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2.741 

5.018 

3-290 

5-854 

3-838 

6.690 

4-386 

7-527 

4-935 

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2.760 

5-003 

3-312 

5-837 

3.864 

6.671 

4.416 

7-505 

4.967 

56^ 

33^ 

2.778 

4.989 

3-333 

5.820 

3-889 

6.652 

4-445 

7-483 

5.000 

56k 

34° 

2.796 

4-974 

3-355 

5-803 

3-W4 

6.632 

4-474 

7.461 

5-033 

56 

34% 

2.814 

4.960 

3-377 

5.786 

3-940 

6.613 

4.502 

7-439 

5-065 

553^ 

2.832 

4-945 

3-398 

5-769 

3-965 

6-593 

4-531 

7-4I7 

5-098 

55^ 

31*. 

2.850 

4-930 

3-420 

5-752 

3-990 

6-573 

4-56o 

7-  3')  5 

5-130 

55X 

35° 

2.868 

4-9x5 

3-441 

5-734 

4.015 

6-553 

4-589 

7-372 

5.162 

55° 

35% 

2.886 

4.900 

3-463 

5-716 

4.040 

6-533 

4.617 

7-350 

5-194 

5434' 

2.904 

4-885 

3-484 

5-699 

4.065 

6.513 

4.646 

7-327 

5.226 

54^ 

353/l 

2.921 

4.869 

3-505 

5-68i 

4.090 

6-493 

4.674 

7-304 

5-258 

54  % 

36C 

2-939 

4-854 

5-663 

4.115 

6.472 

4.702 

7.281 

5.290 

54 

3b% 

2-957 

4-839 

3.548 

4-139 

6.452 

4-730 

7-258 

5.322 

53?4: 

36X 

2.974 

4-823 

3-569 

5.627 

4.164 

6.431 

4-759 

7-235 

5-353 

53^ 

36  34 

2.992 

4.808 

3-590 

5-609 

4.188 

6.410 

4.787 

7.211 

5.385 

3-oog 

4.792 

3.611 

5-590 

4-213 

6.389 

4.815 

7.188 

5.416 

53  °4 

3~% 

3.026 

4.776 

3-632 

5-572 

4-237 

6.368 

4.842 

7.164 

5-448 

52  # 

3-044 

4.760 

3-653 

5-554 

4-261 

6-347 

4.870 

7.140 

5-479 

37% 

3.061 

4-744 

3-673 

5-535 

4-2G6 

6.326 

4.898 

7.116 

5-510 

52* 

38° 

3.078 

4-728 

3-694 

4.310 

6.304 

4-9-5 

7.092 

5-54t 

52  J 

38j^ 

3.095 

4.712 

3-7I5 

5-497 

4-334 

6.283 

4-953 

7.068 

5-572 

51  ^ 

38^ 

3-"3 

4.696 

3-735 

5-478 

4-358 

6.261 

4.980 

7-043 

5-603 

51  y2 

38^ 

3-130 

4.679 

3-  756 

5-459 

4.381 

6.239 

5.007 

7.019 

5-633 

51  1A 

39° 

3-147 

4-663 

3-776 

5-440 

4-405 

6.217 

5-035 

6-994 

5-664 

51° 

§* 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

~w 

Jt 

5 

1 

6 

' 

7 

3 

9 

i 

LATITUDES   AND    DEPARTURES. 


49 


> 

i 

J 

1 

5 

•g) 

. 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

I 

39" 

0.777 

0.629 

•554 

•259 

2-331 

.888 

3.109 

2.5I7 

3.886 

51° 

39V 

0.774 

0.633 

•549 

.265 

2.323 

.898 

3-098 

2-531 

3.872 

SoV 

39^ 

0.772 

0.636 

•543 

.272 

2.315 

.908 

3.086 

2-544 

3.858 

$0% 

39V 

0.769 

0.639 

-538 

•279 

2.307 

.918 

3-075 

2.558 

3-844 

40° 

0.766 

0-643 

•532 

.286 

2.298 

.928 

3.064 

2-571 

3-830 

50° 

4o/4' 

0.763 

0.646 

.526 

.292 

2.290 

•938 

3-053 

2.584 

3-816 

49V 

40*4 

0.760 

0.649 

•521 

.299 

2.281 

.948 

3.042 

2.598 

3.802 

49^ 

4034; 

0.758 

0-653 

•515 

.306 

2.273 

•958 

3-030 

2.611 

3.788 

41° 

0.755 

0.656 

•509 

.312 

2.264 

.968 

3.019 

2.624 

3-774 

49° 

41* 

0.752 

0.659 

•504 

•319 

2.256 

.978 

3-007 

2.637 

3-759 

48v 

0.749 

0.663 

.498 

.325 

2.247 

.988 

2.996 

2.650 

3-745 

48^ 

4i# 

0.746 

0.666 

.492 

•332 

2.238 

.998 

2.984 

2.664 

3-730 

42 

0-743 

0.669 

.486 

.338 

2.229 

2.007 

2-973 

2.677 

3.716 

48°4 

42V 

0.740 

0.672 

.480 

•345 

2.221 

2.017 

2.961 

2.689 

3.701 

47V 

42^ 

0-737 

0.676 

•475 

•351 

2.212 

2.027 

2.949 

2.702 

3.686 

47^ 

42?4: 

0-734 

0.679 

•469 

-358 

2.2O3 

2.036 

2-937 

2.715 

3-672 

47V 

43 

Q-731 

0.682 

•463 

•364 

2.194 

2.046 

2.925 

2.728 

3-657 

41° 

43V 

0.728 

0.685 

•457 

•370 

2.185 

2.056 

2.913 

2.741 

3-642 

46V 

43  K 

0.725 

0.688 

•  451 

•377 

2.176 

2.065 

2.901 

2.753 

3-627 

43  3/ 

0.722 

0.692 

•445 

•383 

2.167 

2.075 

2.889 

2.766 

3.612 

4&V 

44 

0.719 

0.695 

•439 

•389 

2.158 

2.084 

2.877 

2.779 

3-597 

46° 

44V 

0.716 

0.698 

•433 

.396 

2.149 

2.093 

2.865 

2.791 

45  X 

44  ]/2 

0.713 

0.701 

.427 

.402 

2.140 

2.103 

2-853 

2.804 

3-566 

45^ 

44% 

0.710 

0.704 

.420 

.408 

2.I3I 

2.II2 

2.841 

2.816 

3-551 

45V 

45 

0.707 

0.707 

.414 

.414 

2.  121 

2.  121 

2.828 

2.828 

3.536 

45° 

B'ring 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

B'ring 

1 

5 

( 

> 

J 

r 

t 

$ 

< 

> 

•QD 

JL 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Jjj 

39° 

3-147 

4-663 

3-776 

5-440 

4.405 

6.217 

5-035 

6.994 

5-664 

"sF 

39V 

3.164 

4.646 

5.421 

4.429 

6.195 

5.062 

6.970 

5-694 

sov 

39^ 

3-i8o 

4.630 

3^8l6 

5.401 

4-453 

6.173 

5-089 

6-945 

5-725 

50^ 

39V 

3-197 

4.613 

3.837 

4.476 

6.151 

6.920 

5-755 

40° 

3.214 

4.596 

3-857 

5-362 

4-500 

6.128 

5,142 

6.894 

5.785 

50° 

4°V 

3-231 

4-579 

3-877 

5-343 

4-523 

6.106 

5.169 

6.869 

5-815 

49  X 

4°//J 

3-247 

4-562 

5-323 

4-546 

6.083 

5.196 

6.844 

5.845 

49K 

4")  V 

3.264 

4-545 

3-9I7 

5-303 

4-569 

6.061 

5-222 

6.818 

5.875 

41° 

3.280 

4-528 

3-936 

5-283 

4-592 

6.038 

5-248 

6.792 

5-905 

49° 

4TV 

3-297 

4-5ii 

3-956 

5-263 

4.615 

6.015 

5-275 

6.767 

5-934 

48V 

41^ 

3-313 

4.494 

3-976 

5-243 

4-638 

5-992 

5-301 

6.741 

5.964 

48^ 

4134' 

3-329 

4.476 

3-995 

5-222 

4.661 

5-968 

5-327 

6-715 

5-993 

48'^ 

42 

4-459 

4.015 

5-202 

4-684 

5-945 

5-353 

6.688 

6.022 

48° 

42V 

3'  362" 

4-441 

4-034 

5.182 

4./07 

5-922 

5-379 

6.662 

6.051 

47% 

42/^ 

3-37S 

4.424 

4-054 

5.I6I 

4.729 

5-898 

5-405 

6.635 

6.08O 

4T/2 

4234^ 

3-394 

4.406 

4-073 

5.140 

4-752 

5.875 

5-430 

6.609 

6.IO9 

47V 

43" 

3.410 

4-388 

4.092 

5.119 

4-774 

5-851 

5-456 

6.582 

6.138 

4f° 

43V 

3-426 

4-3/0 

4.111 

5-099 

4.796 

5-827 

5,481 

6-555 

6.167 

46V 

43K 

3-442 

4-352 

4.130 

5.078 

4.818 

5-803 

5-507 

6.528 

6.195 

4334: 

3-458 

4-334 

4.149 

5-057 

4.841 

5-779 

5-532 

6.501 

6.224 

4&V 

44J 

3-473 

4.316 

4.168 

5-035 

4.863 

5-755 

5-557 

6.474 

6.252 

46° 

44V 

3-489 

4.298 

4.187 

5.014 

4.885 

5-730 

5.582 

6.447 

6.280 

45V 

44K 

3-505 

4.280 

4.206 

4-993 

4.906 

5-706 

5.607 

6.419 

6.308 

45^ 

3-520 

4.261 

4.224 

4.971 

4.928 

5.681 

5.632 

6.392 

6.336 

45V 

4454 

3-536 

4-243 

4-243 

4-950 

4.950 

5-657 

5.657 

6.364 

6364 

45° 

B'ring 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep 

J..-U. 

B'rinjr 

TABLES   AND    FORMULAS.  51 


TABLE  OF 

HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 

The  formulas  used  in  the  computation  of  the  following 
tables  furnish  expressions  for  horizontal  distances  and 
differences  of  elevation  for  stadia  measurements  with  the 
conditions  that  the  stadia  rod  be  held  vertical  and  the  stadia 
wires  be  equidistant  from  the  center  wire.  The  formulas 
used  are  as  follows:  For  the  horizontal  distance 

D  =  c  cos  n  +  a  k  cos2  «,  (94.)     Art.  1 3O1 . 

in   which  D  =  the    corrected   distance  ;    c  =  the    constant  ; 
a  k  =  the  stadia  distance,  and  n  —  the  vertical  angle. 

For  the  difference  of  elevation,  the  following  formula  is 
used: 

E  =  c  sin  n  +  a  k  ^2f.  (95.)     Art.  1 3O1 . 

For  application  of  tables  see  Art.  13O1* 


TABLES   AND    FORMULAS. 


53 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 


0 

• 

I 

0 

2 

« 

3 

° 

Minutes, 
o'  

2   

4           .          ... 

Hor. 
Dist. 

IOO.OO 
IOO.OO 
IOO.OO 

Diff. 
Elev. 

.00 

.06 

.  12 

Hor. 

Dist. 

99-97 
99-97 

Diff. 
Kiev. 

1.74 
i.  so 

1.86 

Hor. 
Dist. 

99.88 
99.87 

00.87 

Diff. 
Elev. 

3-49 

3-55 
3  60 

Hor. 
Dist. 

99-73 
99.72 
99.71 

Diff. 
Elev. 

5-23 
5.28 

5-34 

6  
8  

10   

12   
14      

IOO.OO 
IOO.OO 
100.00 

IOO.OO 
IOO.OO 

•17 
•23 
.29 

-35 
.41 

99.96 
99.96 
99.96 

99.96 
99-95 

1.92 
1.98 
2.04 

2.09 
2.15 

99.87 

99-86 
99.86 

99-85 
99-85 

3-66 
3-72 
3-78 

3-84 
3-9° 

99.71 
99.70 
99.69 

99.69 
99.68 

5-40 
5-46 

5-52 

5-57 
5.63 

16  
18  

20  
22                              .... 

IOO.OO 
IOO.OO 
IOO.OO 

IOO.OO 

•47 

•52 
•58 

.64 

99-95 
99-95 
99-95 

2.21 
2.27 
2-33 
2.38 

99.84 
99.84 
99-83 

QQ  8s 

3-95 
4.01 
4.07 

4-I3 

99.68 
99.67 
99.66 

99.66 

5-69 

5-75 
5-80 

5-86 

24  
26  
28   

100.00 

99-99 
99-99 

.70 
.76 
.81 

•87 

99-94 

99-94 
99-93 

2-44 
2.50 
2.56 

2  62 

99.82 
99.82 
99.81 

QQ  8  1 

4.18 
4.24 
4-30 
4.36 

99-65 
99.64 
99-63 
09.6^ 

5-92 
5-98 
6.04 
6.09 

32  

34  

99-99 
99-99 

•93 
•99 

99-93 
99-93 

2.67 
2.73 

99.80 
99.80 

4.42 
4.48 

99.62 
99.62 

6.15 

6.21 

36 

1.05 

2.7Q 

4-53 

qq.6l 

6  27 

38                      

99-99 

i.  ii 

99.92 

2.85 

4-59 

90.60 

6-33 

40          

99-99 

1.16 

99.92 

2.gl 

99.78 

4-65 

99-59 

6.38 

42 

1.22 

2.  Q7 

00.78 

4-  71 

99.59 

6.44 

44                       .... 

99-98 

1.28 

99.91 

3-O2 

00.77 

4.76 

99.58 

6.50 

46    

99.98 

1.34 

99.90 

3.08 

00.77 

4.82 

99-57 

6.56 

48  
50  

99.98 
99.98 

1.40 

i-45 

99.90 
99.90 

3-14 
3-20 

99.76 
99.76 

4.88 
4-94 

99-56 
99.56 

6.6r 
6.67 

52  
54  
56  

99.98 
99.98 
99-97 

i-5i 
i-57 
1.63 

99.89 
99.89 
99.89 

3.26 
3-31 
3-37 

99-75 
99-74 
99.74 

4-99 
5-05 
5.11 

99-55 
99-54 
99-53 

6-73 
6.78 
6.84 

58  
60  

99-97 
99-97 

1.69 
1.74 

99.88 
99.88 

3-43 
3-49 

99-73 
99-73 

5-i7 
5-23 

99-52 
99-51 

6.90 
6.96 

c=    -75  

•75 

.01 

•75 

.02 

•  75 

•03 

•75 

•05 

C  =  1.  00   

1.  00 

.01 

I.OO 

•03 

I.OO 

.04 

I.OO 

.06 

c  =  1.25  

1.25 

.02 

1-25 

•03 

1.25 

•05 

1-25 

.08 

TABLES   AND    FORMULAS. 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 


4C 

> 

5 

3 

6 

) 

7 

3 

Minutes. 

Hor. 
Dist. 

99.51 

Diff, 
Elev. 
6.96 

Hor. 
Dist. 

99.24 

Diff. 
Elev. 

8.68 

Hor. 
Dist. 
98.91 

Diff. 
Elev. 
10.40 

Hor. 
Dist. 

98.51 

Diff. 
Elev. 

12.10 

2 

99-51 

7.02 

99.23 

8.74 

98.90 

10.45 

98.50 

12.15 

99-  50 

7.07 

99.22 

8.80 

98.88 

10.51 

98.48 

12.21 

6  
8  

99.49 
99.48 
99-47 

7-13 
7.19 

7.25 

99-21 
99.20 
99-  *9 

8.85 
8.91 
8.97 

98.87 
98.86 
98.85 

10.57 
10.62 

10.68 

98.47 
98.46 
98.44 

12.26 
12.32 
12.38 

12  
14  

16 

99.46 
99.46 
99-45 

7-30 
7-36 
7.42 

99.18 
99.17 
99.16 

9-03 
9.08 

Q.  14 

98.83 
98.82 
98.81 

10.74 
10.79 
10.85 

98-43 
98.41 
98.40 

12.43 

12.49 

I2o5 

18 

99.44 

7.48 

99.15 

g.2O 

98.80 

10.91 

98.39 

12.  60 

20  

22  
24                         

99-43 
99.42 
99.41 

7-53 

7-59 
7.65 

99.14 

99-13 
99.11 

9-25 
9-31 

9-37 

98.78 

98.77 
98.76 

10.96 

11.02 
II.08 

98-37 

98.36 
98.34 

12.66 

12.72 
12.77 

26                

99.40 

7.71 

99.10 

9-43 

98.74 

11.13 

98.33 

12.83 

28 

99-39 

7.76 

Q  48 

98.73 

Il.lg 

98.31 

12.88 

30  
32                 

99-38 
99.38 

7.82 
7.88 

99.08 
99.07 

9-54 
9.60 

98.72 
98.71 

11.25 
11.30 

98.29 
93.28 

12.94 
13-00 

99-37 

7-94 

99.06 

0.65 

98.69 

11.36 

98.27 

13.05 

16 

99.36 

7-99 

99.05 

9.71 

98.68 

11.42 

98.25 

13.11 

*8 

99.35 

8-05 

99.04 

9-77 

98.67 

11.47 

98.24 

I3-I7 

40                

99-34 

8.ii 

99.03 

9.83 

98.65 

ii-53 

98.22 

13.22 

42 

99-33 

8.17 

99.01 

9.88 

98.64 

11.59 

98.20 

13-28 

44          

99.32 

8.22 

99.00 

9.94 

98.63 

11.64 

98.19 

*3  33 

46  

99.31 

8.28 

98.99 

IO.OO 

98.61 

11.70 

98.17 

I3-39 

48  

5° 

99-30 
99.29 

8-34 
8.40 

98.98 
98.97 

10.05 

IO.  II 

98.60 

98.58 

11.76 
11.81 

98.16 
98.14 

13-45 
I3-50 

52  

99.28 

8.45 

98.96 

10.17 

98.57 

11.87 

98.13 

I3-56 

54 

99.27 

8.51 

98.94 

IO.  22 

98.56 

11.93 

98.11 

i',6i 

56  
58    

99.26 
99-25 

8-57 
8.63 

98.93 
98.92 

10.28 
IO.34 

98-54 
98.53 

11.98 
12.04 

98.10 
98.08 

13-67 
13-73 

60 

8.68 

98.91 

IO.4O 

98.51 

12.10 

98.06 

I3-78 

c=   .75  

•75 

.06 

•75 

.07 

•75 

.08 

•74 

.10 

c  =  i.oo  

I.OO 

.08 

•99 

.09 

•  99 

.11 

•99 

•13 

c  =  1.25  

1-25 

.10 

1.24 

.11 

1.24 

.14 

1.24 

.16 

TABLES   AND    FORMULAS. 


55 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 


8 

3 

9 

3 

1C 

,° 

II 

0 

Minutes. 

Hor. 
Dist. 

98.06 

Diff. 
Elev. 

13.78 

Hor. 
Dist. 

97-55 

Diff. 
Elev. 

15.45 

Hor. 
Dist. 
96.98 

Diff. 
Elev. 

17.10 

Hor. 
Dist. 

96.36 

Dili: 
Elev. 

18  7*? 

2           ...          

98.05 

13.84 

97-53 

15-  51 

96.96 

17.  16 

96.34 

18.78 

4  
6  
8  

98-03 
98.01 
98.00 

13.89 
13-95 
14.01 

97-52 
97-50 
97.48 

15-56 
15.62 
15-67 

96-94 
96.92 
96.90 

17.21 
17.26 
17.32 

96-32 
96.29 
96.27 

18.84 
18.89 

18.95 

97.98 

14.06 

97.46 

15.73 

96.88 

17-37 

96.25 

12  

97-97 
97-95 

14.12 
14.17 

97-44 
97-43 

15.78 
15.84 

96.86 
06.84 

17-43 
17  48 

96.23 
96.21 

19.05 

16 

97-93 

14.23 

97.41 

15.89 

96.82 

17.54 

96.  1  8 

io  16 

18  
20  

97.92 
97.90 

14.28 

14-34 

97-39 
97-37 

15-95 
16.00 

96.80 
96.78 

17-59 
17-65 

96.16 
96.14 

19.21 
19.27 

22                     

97.88 

14.40 

97-35 

1  6.  06 

96.76 

17.70 

96.12 

iq  32 

24        

97.87 

14.45 

97-33 

1  6.  ii 

96.74 

17.76 

96.09 

19.38 

26  

97.85 

14.51 

97.31 

16.17 

96.72 

17.81 

96.07 

19-43 

28 

97.83 

14  56 

97.29 

l6  22 

06  70 

17.86 

96.05 

IQ  48 

1O 

97.82 

14.62 

97.28 

16.28 

06.68 

17.92 

96.03 

IQ  54 

32  

•34 

97.80 
97.78 

14.67 
14.73 

97.26 
97.24 

16-33 

16  39 

96.66 
96.64 

17-97 
18.03 

96.00 
05.08 

19.59 

36               

97.76 

14.79 

97.22 

16.44 

96.62 

18.08 

95-96 

IQ.  7O 

38             

97-75 

14.84 

97.20 

16.50 

96.60 

18.14 

95-93 

19.75 

40  
42           ... 

97-73 
97.71 

14.90 
14.95 

97.18 
97.16 

16-55 
16.61 

96-57 

Q6.  55 

18.19 
18.24 

95-91 
95.89 

Ig.SO 
lg.86 

44  

97.69 

15.01 

97.14 

1  6.  66 

96.  53 

18.30 

95-86 

ig.gl 

46  

48 

97.68 
07.66 

15.06 
1512 

97.12 
97-  Io 

16.72 
16  77 

96.51 

18.35 
18.41 

95.84 
95.82 

19.96 
2O  O2 

50 

97.64 

15-17 

97.08 

16.83 

96.47 

18.46 

95-79 

2O.O7 

52  

97.62 

15.23 

97.06 

16.88 

96.45 

18.51 

95-77 

20.  1  2 

54 

97  61 

15.28 

97.04 

1  6  04 

06.42 

18.57 

95-  75 

20.  1  8 

56                  .... 

97-59 

15.34 

97.02 

16.99 

96.40 

18.62 

95.72 

20.23 

58  

97-  57 

15.40 

97.00 

17.05 

96.38 

18.68 

95.70 

20.28 

60  

97-55 

15.45 

96.98 

17.10 

96.36 

18.73 

95-68 

20.34 

C=    .75  

•74 

.11 

•74 

.12 

•74 

.14 

•73 

•  15 

C  =  I.OO   

•99 

•  15 

•99 

.16 

.98 

.18 

.98 

.20 

C  =  1.25  

1.23 

.18 

1-23 

.21 

1-23 

•23 

1.22 

•25 

TABLES   AND    FORMULAS. 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 


12 

0 

i; 

° 

I/ 

\° 

1C 

Minutes. 

o' 

Hor. 
Dist. 

95.68 

Diff. 
Elev. 

20.34 

Hor. 
Dist. 

Diff. 
Elev. 

Hor. 
Dist. 

04  15 

Diff. 

Elev. 

21  47 

Hor. 
Dist. 

QO     OQ 

Diff. 
Elev. 

2 

95.65 

20.39 

04.0! 

Q4.  12 

23.52 

Q3.27 

25  05 

4  
6  
8  

95-63 
95.61 
95-58 

20.44 

20.50 
20.55 

94.89 
94.86 
94.84 

22.02 
22.O8 
22.13 

94.09 
94.07 
94.04 

23.58 
23-63 
23.68 

93-24 
93-21 
93.18 

25.10 
25-15 
25.20 

10 

95.56 

20.60 

04.81 

04.01 

23  73 

93-  *6 

25  25 

12  

95.53 

20.66 

94-  79 

22  2^ 

93.98 

23.78 

93-13 

25.30 

14  

95.51 

20.71 

94.76 

22.28 

93-95 

23.83 

93.10 

25.35 

16 

95-49 

20.76 

94-  73 

Q3    Q-l 

23.88 

93-07 

25  40 

18 

95.46 

20.81 

94-  71 

93-  QO 

21-03 

93.04 

25.45 

20  
22 

95-44 
95.41 

20.87 
20  92 

94.68 
94.66 

22.44 

93.87 

QQ     84 

23-99 

93-oi 
92.98 

25-50 

25  55 

24  
26  
28  

95-39 
95-36 

95-34 
05-32 

20.97 
21.03 
21.08 

21   11 

94-63 
94.60 
94-58 

22.54 
22.60 
22.65 

93.81 
93-79 
93-76 

24.09 
24.14 
24.19 
24  24 

92.95 
92.92 
92.89 

02.86 

25.60 
25-65 
25.70 

25  75 

32                 

95-29 

21.  l8 

94.  52 

22  75 

93-7° 

24.29 

92.83 

25.80 

34  
36  

95-27 
95.24 

21.24 
21.29 

94-50 
O4.47 

22.80 
22.85 

93-67 
93.65 

24-34 
24.39 

92.80 
92.77 

25.85 
25.90 

38 

95-22 

21-34 

93.62 

24.44 

92.74 

2G  05 

40  

42  
44 

95-19 
95-17 

95.14 

21-39 

21-45 
21.50 

94.42 

94-39 
94-36 

22.96 
23.01 

93-59 
93-56 
93-53 

24-49 

24-55 
24.60 

92.71 

92.68 
92.65 

26.OO 

26.05 

26.  10 

46  
48  
50 

95.12 
95-09 

05.07 

21-55 
21.60 
21.66 

94-34 
94-31 
94.28 

23.11 

23.16 

93-50 
93-47 
03  45 

24-65 
24.70 

24.  75 

92.62 

92-59 
92.56 

26.15 
26.20 
26  25 

52  

54 

95.04 

05  O2 

21.71 
21.76 

94.26 

23.27 

93-42 

24.80 

24.85 

92.53 

26.30 

56  

94-99 

21.  8l 

94.20 

23.37 

93.36 

24.90 

92.46 

26.40 

58    
60  

94-97 
94-94 

21.87 
21.92 

94.17 
94-15 

23-42 

23-47 

93-33 
93-30 

24-95 
25.00 

92-43 
92.40 

26.45 

26.50 

c=  .75  

•73 

.16 

•73 

•  17 

•73 

.19 

•72 

.20 

C  —  i.oo 

.98 

.22 

•97 

•97 

.25 

.06 

•27 

c  =  1.25  

1.22 

.2? 

1.  21 

.29 

I.  21 

•3i 

I.  2O 

•34 

TABLES   AND    FORMULAS. 


57 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION   FOR  STADIA   MEASUREMENTS. 


id 

0 

*7 

0 

i£ 

0 

*S 

Minutes. 

o'  
2  

4  

6 

Hor. 
Dist. 

92.40 
92-37 
92-34 
92.31 

Diff, 
Elev. 

26.50 
26.55 
26.59 
26.64 

Hor. 
Dist. 

91-45 
91.42 

91-39 
91.35 

Diff. 
Elev. 

27.96 
28.01 
28.06 
28.10 

Hor. 
Dist. 

90.45 
90.42 
90.38 
90.35 

Diff. 
Elev. 

29-39 
29.44 
29.48 
29.53 

Hor. 
Dist. 

89.40 
89.36 
89-33 
89.29 

Diff. 
Elev. 

30.78 
30-83 
30.87 
30.92 

3  

92.28 
92.25 

26.69 
26.74 

91.32 
91.29 

28.15 
28.20 

90.31 
90.28 

29.58 
29.62 

89.26 
89.22 

30.97 
31.01 

12  

U  

16 

92.22 
92.19 
92.15 

26.79 
26.84 
26.89 

91.26 
91.22 

91.19 

28.25 
28.30 

28.34 

90.24 
90.21 
90.18 

29.67 
29.72 
29.76 

89.18 
89.15 
89.11 

31.06 
31.10 
31.  15 

18  

2O  

92.12 
92.09 

26.94 
26.99 

91.16 
91.12 

28.39 
28.44 

90.14 
90.  1  1 

29.81 
29.86 

89.08 
89.04 

31-19 
31.24 

92.06 

27.04 

91.09 

28.49 

90.07 

29.90 

89.00 

31  28 

24  

92.03 

27.09 

91.06 

28.54 

90.04 

29.95 

88.96 

31  33 

26  

92.00 

27.13 

91.02 

28.58 

90.00 

30.00 

88.93 

31.38 

28 

91.97 

27.18 

90.99 

28.63 

89.97 

30.04 

88  80 

91-93 

27.23 

90.96 

28.68 

89.93 

30.09 

88.86 

91.90 

27.28 

90.92 

28.73 

89  90 

30.  14 

88  82 

34  
36  
38  
40  

42  
44  
46  
48  

91.87 
91.84 
91.81 
91.77 

91-74 
91.71 
91.68 
91.65 
91.61 

27-33 
27-38 
27-43 
27-48 

27-52 
27-57 
27.62 
27.67 

90.89 
90.86 
90.82 
90.79 

90.76 
90.72 
90.69 
90.66 
90.62 

28.77 
28.82 
28.87 
28.92 

28.96 
29.01 
29.06 
29.11 

89.86 
89-83 
89.79 
89.76 

89.72 
89.69 
89-65 
89.61 

30.19 
30.23 
30.28 
30.32 

30.37 
30.41 
30.46 
30.51 

88.78 
88-75 
88.71 
88.67 

88.64 
88.60 

88.56 
88-53 
88  49 

3I-56 
31.60 
31-65 
31.69 

31-74 
31-78 
31-83 
31-87 

52  
54  
56  

58 

91.58 
91-55 
91.52 
91.48 

27-77 
27.81 
27.86 

90-59 

9°-55 
90-52 
90.48 

29.20 
29.25 
29.30 

89-54 
89.51 
89.47 

80  44 

30.60 
30-65 
30.69 

30.  74 

88.45 
88.41 
88.38 
88  14 

31.96 
32.01 
32-05 

60  

91.45 

27.06 

90.45 

29.39 

Su  40 

30.78 

88  30 

c=  -75  

.72 

.21 

•72 

•23 

•71 

.24 

•7i 

.25 

C  =  1.  00  

.86 

.28 

•  -95 

•30 

•95 

•32 

•94 

•33 

c  =  1.25  

1.20 

•35 

1.19 

•38 

|    1-19 

.40 

1.18 

.42 

58 


TABLES   AND    FORMULAS. 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS, 


2C 

0 

21 

0 

22 

o 

23 

Minutes. 

o'  
2  

Hor. 
Dist. 

88.30 
88.26 

Diff. 
Elev. 
32.14 

32.18 

j 

Hor. 
Dist. 

87.16 
87.12 

Diff. 
Elev. 
33-46 

33-  5° 

Hor. 
Dist. 

85-97 
85.93 

Diff. 
Elev. 

34-73 
34.77 

Hor. 
Dist. 

84-73 
84.69 

Diff. 
Elev. 

35-97 
36.01 

88.23 

32.23 

87.08 

33  54 

85.89 

34.82 

84.65 

36.05 

6 

88.19 

32.27 

87.04 

33.  59 

85.85 

-24.86 

84  6  1 

36.09 

8         

88.15 

32.32 

87.00 

33.63 

85.80 

34.90 

84.  57 

36.  13 

10      

88.11 

32.36 

86.96 

33-67 

85.76 

34-94 

84.52 

36.17 

12                            

88.08 

32.41 

86.92 

33  72 

85.72 

34.98 

84.48 

36.21 

14  

16  
18  

86.04 
88.00 
87-96 
87.93 

32.45 
32-49 
32-54 
32.58 

86.88 
86.84 
86.80 
86.77 

33-76 
33-8o 
33-84 
TV  80 

85.68 
85-64 
85.60 
85.56 

35.02 
35-07 
35-n 
35-  *5 

84.44 
84.40 
84-35 
84  ii 

36.25 
36.29 
36.33 
36  37 

22           

87.89 

32.63 

86.73 

33-93 

85.52 

35.19 

84.27 

36.41 

24        

87.85 

32.67 

86.69 

33-97 

85.48 

35.23 

84.23 

36-45 

26 

87.81 

32.72 

86.65 

34.01 

85.44 

35.27 

84  18 

36  49 

28 

87.77 

32.76 

86.  6  1 

34.06 

85.40 

35.31 

84  14 

36  53 

87.74 

32.80 

86.57 

34-  Io 

85.36 

35.36 

84.10 

36.  57 

32  

34  
36  
38            

87.70 
87.66 
87.62 
87.58 

32.85 
32.89 
32-93 
32.98 

86.53 
86.49 
86.45 
86.41 

34-M 

34.18 
34-23 
34.27 

85-31 

85-27 
85.23 
85.19 

35-40 
35-44 

35-48 
35.52 

84.06 
84.01 

83-97 
83.93 

36.61 
36.65 
36-69 
36.73 

40  

42  
44  
46  
48 

87-54 

87-51 
87.47 
87-43 
87.39 

33-02 

33-0? 
33-" 
33-15 
33.20 

86.37 

86-33 
86.29 
86.25 
86.21 

34-31 

34-35 
34-40 
34-44 

14  48 

85-15 

85.11 
85.07 
85.02 
84.98 

35-56 
35-6o 
35.64 
35-68 
35.72 

83-89 

83-84 
83.80 
83.76 
83.  72 

36.77 
36.80 
36-84 
36.88 
36.92 

50  

52  
54        

87-35 

87-31 
87.27 

33-24 
33-28 
33-33 

86.17 

86.13 
86.09 

34-52 

34-57 
34.61 

84-94 
84.90 
84.86 

35.76 
35.80 
35.85 

83.67 

83-63 
83.59 

36.96 
37-00 
37-°4 

56  

87.24 

33-37 

86.05 

34.65 

84.82 

35-Sg 

83.54 

37.08 

58    

87.20 

33.41 

86.  o  i 

34-  69 

84.77 

35-93 

83.50 

37.12 

60 

87.16 

33-46 

85.97 

34  73 

84.73 

35-97 

83.46 

37.16 

c=    .75  

.70 

.26 

.70 

.27 

.69 

.29 

.69 

•30 

C  =  I.OO  

•94 

•35 

•93 

•37 

.92 

•38 

.92 

.40 

c  —  1.25  . 

i  16 

46 

48 

TABLES   AND    FORMULAS. 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 


24 

° 

25 

2( 

)° 

27 

0 

Minutes, 
o'  

2   

Hor. 
Dlst, 

83-46 
83.41 

Diff. 
Elev. 

37.16 
37.20 

Hor. 
Dist. 

82.14 
82.09 

Diff, 

Elev. 

38.30 
38.34 

Hor, 
Dist. 

80.78 
80.74 

Diff. 
Elev, 

39-40 
39.44 

Hor. 
Dist. 

79-39 
79.34 

Diff. 
EleY. 

40.45 
40.49 

83.37 

37.23 

82.05 

38.38 

80.69 

39-47 

79-3O 

40.  52 

6  
8          

83-33 
83.28 

37-27 
37.31 

82.01 
81.96 

38.41 
38.45 

80.65 
80.60 

39-51 
39-  54 

79-25 
79.20 

40-55 
40.59 

10  

12 

83.24 
83.20 

37-35 
37.39 

81.92 

81.87 

38-49 
38.53 

80.55 
80.51 

39-58 
39.61 

79-15 
79.11 

40.62 
40.66 

14  

16  

18 

83-15 
83.11 
83.07 

37-43 
37-47 
•77  e.i 

81.83 
81.78 
81.74 

38.56 
38.60 
38  64 

80.46 

80.41 
80.37 

39-65 
39-69 
39-72 

79.06 
79-oi 
78.96 

40.69 
40.72 
40.76 

20 

83.02 

37-54 

81.69 

38.67 

80.32 

39-  76 

78.92 

40.79 

22      

82.98 

37.58 

81.65 

38.71 

80.28 

39-79 

78.87 

40.82 

24  

82.93 

37.62 

81.60 

38.75 

80.23 

39.83 

78.82 

40.86 

26 

82.89 

37.66 

81.56 

38.78 

80.  1  8 

39.86 

78.77 

40.89 

28                           

82.85 

37.70 

81.51 

38.82 

80.14 

39-90 

78.73 

40.92 

30      

82.80 

37.74 

81.47 

38.86 

80.09 

39-93 

78.68 

40.96 

72 

82.76 

37-77 

81.42 

38.89 

80.04 

30.07 

78.63 

40.99 

34 

82.72 

37.81 

81.38 

38.93 

80.00 

40.00 

78.58 

41.02 

36  
38  

82.67 
82.63 
82  58 

37.85 
37-89 

8i-33 
81.28 
81.24 

38.97 
39.00 

79-95 
79.90 
79.86 

40.04 
40.07 

78.54 
78.49 
78.44 

41.06 
41.09 
41.12 

42  
44  
46  

82.54 
82.49 
82.45 

37-96 
38.00 
38.04 

81.19 
81.15 
81.10 

39.08 

39-n 
39.15 

79.81 
79.76 
79.72 

40.14 
40.18 
40.21 

78.39 
78.34 
78.30 

41.16 
41.1.9 
41.22 

48 

82.41 

38  08 

81.06 

30.  1  8 

79.67 

40.24 

78.25 

41.26 

50  

52 

82.36 
82.32 

38.11 
38  15 

81.01 
80.97 

39-22 
39.26 

79.62 
70  58 

40.28 
40.31 

78.20 
78  15 

41.29 
41.32 

54  
56  
58  
60 

82.27 
82.23 
82.18 
82  14 

38.19 
38-23 
38.26 

Og    OQ 

80.92 

80.87 
80.83 
80.78 

39-29 
39-33 
39-36 

79-53 
79.48 
79-44 
70-30 

40-35 
40.38 
40.42 
40.45 

78.10 
78.06 

78.01 

41-35 
41-39 
41.42 
41.45 

c  —    .75 

.68 

aj 

.68 

•32 

.67 

•33 

.66 

•35 

c  =  i.oo  

.91 

.41 

.90 

•43 

.89 

•45 

.89 

.46 

c  —  1.25  

1.14 

•  52 

I-I3 

•54 

1.  12 

•56 

i.  ii 

•58 

TABLES   AND    FORMULAS. 


HORIZONTAL  DISTANCES  AND  DIFFERENCES 
OF  ELEVATION  FOR  STADIA  MEASUREMENTS. 


28 

0 

29 

0 

3^ 

Minutes. 

Hor. 
Dist. 

77.96 

Diff. 
Elev. 

41-45 

Hor. 
Dist. 

76.50 

Diff. 
Elev. 

Hor. 
Dist. 

Diff. 
Elev. 

43  3O 

2  

77.91 

41.48 

76.45 

42.43 

74-95 

43-33 

77.86 

41.  52 

76.40 

42  46 

74.90 

43.36 

6  
8  

77.81 

77-77 
77.72 

41-55 
41.58 
41.61 

76-35 
76.30 
76.25 

42-49 
42-53 

74-85 
74.80 
74-75 

43-39 
43-42 
43-45 

12                            

77.67 

41.65 

76.20 

42.  50 

74.70 

43-47 

14  
16        

77.62 
77-57 

41.68 
41.71 

76-15 
76.10 

42.62 

42.65 

74-65 
74.60 

43-50 
43-53 

18  

77-52 
77.48 

41.74 
41.77 

76.05 
76.00 

42.68 

74-55 
74-49 

43.56 
43-  59 

77.42 

41.81 

75-95 

74-44 

43  62 

77.38 

41.84 

75-9O 

74-39 

43.65 

26                                             

77-33 

41.87 

75.85 

42  80 

74-34 

43.67 

28                                     

77.28 

41.90 

75.80 

4°  8^ 

74.29 

43-7° 

30    

77-23 
77.18 

41-93 
41.97 

75-75 
75.70 

42.86 

42  80 

74-24 
74.19 

43-73 
43.76 

77.13 

42.00 

75-65 

42  Q2 

74.14 

43-79 

36  
38  
40  

42  

77.09 
77.04 
76.99 

76.94 

42-03 
42.06 
42.09 

42.12 

75.60 

75-55 
75-50 

75-45 

42.95 
42.98 

43-oi 
43.04 

74.09 
74.04 
73-99 
73-93 

43.82 
43-84 
43-87 
43.90 

44  

76.89 

42.15 

75-4O 

43-07 

73.88 

43-93 

46  
48  
50            .        ... 

76.84 
76.79 
76.74 

42.19 
42.22 
42.25 

75-35 
75-30 
75.25 

43.10 
43-13 

43-  !6 

73-83 
73.78 
73-73 

43-95 
43-gS 
44.01 

52 

76.69 

42.28 

75.20 

4T  18 

73-68 

44.04 

54  
56  
58    

76.64 

76.59 
76.55 

42.31 

42-34 
42.37 

75-15 
75-io 
75.05 

43.21 
43-24 
43.27 

73-63 

73-58 
73.53 

44.07 
44.09 
44.12 

60  

76.50 

42.40 

75.00 

43-30 

73-47 

44-15 

c=    -75  

.66 

•36 

•65 

•37 

•  65 

•38 

c  =  i.  oo  

.88 

.48 

.87 

•49 

.86 

•  51 

c  =  1.25  

1.  10 

.60 

1.09 

.62 

1.08 

.64 

TABLES  AND   FORMULAS.  61 


TABLE  OF 

RADII  AND  CHORD  AND  TANGENT 
DEFLECTIONS. 

The  formulas  used  in  the  computation  of  the  following 
tables  are  as  follows: 

For  Radii,  R  =  Jj^jy.         (89.)     Art.  1249. 
For  Chord  Deflections, 

d=^>.         (92.)     Art.  1255. 
For  Tangent  Deflections, 

tan  deflection  =  £--        (93.)    Art.  1255. 


TABLES   AND    FORMULAS.  63 

TABLE  OF   RADII  AND  DEFLECTIONS. 


Tan- 

Tan- 

Tan- 

De- 

Radii. 

Chord 
Deflec- 

gent 
De- 

De- 

Radii. 

Chord 
Deflec- 

gent 
De- 

De- 

Radii. 

Chord 
Deflec- 

•s? 

gree. 

tion. 

flec- 

^ree. 

tion. 

flec- 

jree. 

tion. 

flec- 

tion. 

tion. 

tion. 

o     5 

68754.94 

•MS 

•°73 

5  15 

091.73 

9.  160 

4.580 

10  50 

529.67 

18.880 

9.440 

10 

34377-48 

.291 

.145 

20 

074-68 

9-305 

•  653 

15 

20 

17188.76 

.582 

.291 

30 

042.14 

9-596 

.798 

513-91 

19.  169 
19.459 

9-585 
9.729 

25 

13751.02 

.727 

•364 

35 

O26.6O 

9.741 

.870 

20 

506-38 

19-748 

9.874 

3° 

11459.19 

•873 

.436 

40 

OII.5I 

9.886 

•943 

30 

499.06 

20.038 

ro.oig 

35 

9822.18 

•509 

45 

^6.87 

10.031 

.016 

40 

491.96 

20.327 

10.164 

40 

8594.41 

1:164 

.582 

50 

Q82.64 

10.177 

.088 

50 

485-05 

20.616 

10.308 

45 

7639-49 

1.309 

.654 

55 

068.  8r 

10.322 

.161 

5° 

6875-55 

1-454 

.727 

12      0 

478-34 

20.906 

10.453 

55 

6250.51 

i.  600 

.800 

6    o 

955-37 

10.467 

•234 

10 

471.8! 

21.195 

10.597 

5 

942.29 

10.612 

.306 

2O 

465-46 

21.484 

10.742 

'1       O 

5729.65 

1-745 

•873 

10 

929-57 

10.758 

•379 

3° 

459.28 

21.773 

,0.887 

5 

5288.92 

1.891 

•945 

15 

917.19 

10.903 

.451 

40 

453-26 

22.063 

11.031 

4911.15 

2.036 

.018 

905-13 

11.048 

•524 

5» 

447.40 

22.352 

11.176 

15 

4583-75 

2.182 

.091 

25 

893-39 

11.193 

•597 

20 

4297.28 

2.327 

.164 

3° 

881.95 

"•339 

.669 

13    o 

441.68 

22.641 

11.320 

25 

4044.51 

2.472 

.236 

35 

870.79 

11.484 

.742 

436.12 

22.  930 

11.465 

3° 

3819.83 

XfiTQ     Q0 

n     -6-j 

.309 

40 

859.92 

11.629 

20 

430.69 

23.219 

11.609 

35 
40 

3OIO.OO 
3437.87 

2.7O3 
2.900 

•454 

5° 

838.97 

11.910 

.960 

40 

420.23 

23  •  796 

".$ 

45 

3274-I7 

3-054 

.527 

55 

828.88 

12.065 

.032 

5° 

415-19 

24.085 

12.043 

5° 

3125.36 

3.200 

.600 

55 

29    9.4 

3-345 

-673 

5 

809.40 

'2.355 

.177 

405-47 

24-663 

12.331 

2      O 

2864.93 

3.490 

•745 

10 

800.00 

12.500 

.250 

2O 

400.78 

24.95I 

12.476 

5 

2750-35 

3-636 

15 

790.81 

12.645 

•323 

3° 

396.20 

25.240 

12.620 

10 

15 

2644.58 
2546.64 

3-781 

3-927 

'.891 
.963 

25 

781.84 
773.07 

12.790 
12.936 

'.III 

50 

391-72 
387-34 

25.817 

2455.70 

4.072 

-036 

3° 

764.49 

I3.o8i 

•  54° 

25 

2371.04 

4.218 

35 

756.10 

13.226 

6.613 

15     o 

383-06 

26.  105 

13-053 

3° 

2292.01 

4-363 

.  181 

40 

747.89 

13-371 

6.685 

IO 

378.88 

26.394 

'3-197 

35 

2218.09 

4.508 

•254 

45 

739.86 

13-516 

6.758 

20 

374-79 

26.682 

'3-341 

40 

2148.79 

4-654 

•327 

50 

732.01 

13.661 

6.831 

30 

370.78 

26.970 

13-485 

45 

2083.68 

4-799 

55 

724-31 

13.806 

6.90; 

4° 

366.86 

27.258 

13.629 

5° 

2022.41 

4-945 

•472 

50 

363-02 

27-547 

13-773 

55 

1964.64 

5.090 

•545 

8     o 

716.78 

i3-95i 

6.976 

5 

709.40 

14.096 

7.048 

16    o 

359-26 

27.835 

I3-9I7 

3     ° 

1910.08 

5-235 

.618 

10 

702.18 

14.241 

IO 

355-59 

28.123 

14.061 

5 

1858.47 

5-38i 

.690 

15 

695.09 

14-387 

•193 

20 

35L98 

28.4II 

14.205 

1809.57 

5.526 

•763 

688.16 

14-532 

.266 

30 

348.45 

28.699 

28        Q86 

14-349 

*5 

1703.  is 

1719.12 

5-817 

.908 

3° 

674.69 

14.822 

.411 

50 

344-99 
341.60 

29.274 

'4-493 
14-637 

25 

1677.20 

5.962 

.981 

35 

668.15 

14.967 

•483 

3° 

1637.28 

6.108 

•°54 

40 

661.74 

15-112 

•556 

I7       0 

338.27 

29.562 

20  .  850 

14.781 

35 
40 

1562.88 

6.398 

.199 

5° 

649.27 

15.402 

.701 

20 

331.82 

30-137 

15-060 

45 

I528.I6 

6-544 

.272 

55 

643-22 

iS-547 

•773 

3° 

328.68 

30-425 

15.212 

5° 

1494.95 

6.689 

•345 

40 

325.60 

30.712 

I5-356 

55 

1463.16 

6.835 

.417 

9    o 

637.27 

15-692 

.846 

5° 

322.59 

31.000 

15-500 

5 

631.44 

15-837 

.918 

4    ° 

1432.69 

6.980 

.490 

625.7! 

15-982 

7.991 

18    o 

319.62 

31.287 

15.643 

5 

1403.46 

7.125 

•563 

15 

620.09 

16.127 

8.063 

IO 

316.71 

31-574 

15.787 

10 

I375-4° 

7.271 

•635 

20 

614-56 

16.272 

8.13) 

20 

313-86 

31.861 

15.931 

15 

1348.45 

7.416 

.708 

25 

609.14 

16.417 

30 

311.06 

32-149 

16.074 

20 

1322.53 

7.56! 

.781 

30 

603.80 

16.562 

8.281 

4° 

308.30 

32-436 

16.218 

25 

1297.58 

7.707 

•853 

35 

598.57 

16.707 

8-353 

5° 

305.60 

32-723 

16.361 

3° 

'273-57 

7-852 

.926 

40 

593.42 

16.852 

8.426 

35 

1250.42 

7-997 

•999 

45 

588.36 

16.996 

8.498 

19    o 

302.94 

33-010 

16.505 

1228.11 

8.143 

.071 

5° 

583-38 

17.141 

8-571 

300.33 

33-296 

16.648 

45 

1206.57 

8.288 

.144 

55 

578.49 

17.286 

8.643 

20 

297.77 

33.583 

16.792 

5° 

1185.78 

8-433 

30 

295-25 

33-870 

i6.935 

55 

1165.70 

8-579 

[289 

10    o 

573.69 

17-431 

8.716 

4° 

292.77 

34-157 

,7-078 

IO 

564-31 

17.721 

8.860 

5° 

290.33 

34-443 

17.222 

5    o 

1146.28 

8  724 

•362 

20 

555-23 

18.011 

9.005 

5 

1127.50 

s.seg 

•435 

30 

546.44 

i8.3«> 

9.150 

20       0 

287.94 

34-730 

17-365 

10 

1109.33 

9.014 

•507 

40 

537-92 

18.500 

9-295 

64 


TABLES   AND    FORMULAS. 
MOMENTS   OF   INERTIA. 


Dotted  Line  Shows  Position 
of  Neutral  Axis. 


td+t.b 


bd  -  *, 


/,/    af-csf, 


TABLES   AND    FORMULAS.  65 

BENDING  MOMENTS  AND  DEFLECTIONS. 


Manner  of  Supporting 
Beams. 


Maximum 

Bending 

Moment,  J/. 


Maximum 
Deflection,  ^. 


Remarks. 


OQOQQQOQOOO 


Cantilever,  more  than 
one  load. 


Cantilever,  uniform 
load  w  Ib.  per  unit 
of  length. 


48  El 


Cantilever,  load  partly 
uniform,  partly  con- 
centrated. 


Simple  beam,  load  at 
middle. 


Simple  beam,  load  at 
some  other  point 
than  the  middle. 


Simple      beam,      uni- 
formly loaded. 


One  end  fixed,  other 
end  supported,  load 
in  the  middle. 


One  end  fixed,  other 
end  supported,  uni- 
formly loaded. 


Both  ends  fixed,  load 

in  the  middle. 


Both   ends   fixed,   uni- 
formly loaded. 


TABLES   AND    FORMULAS. 


SPECIFIC  GRAVITIES  AND  WEIGHTS 
PER   CUBIC    FOOT. 


METALS. 


Substance. 

Specific 
Gravity. 

Weight  per 
Cubic  Foot 
in  Pounds. 

Osmium  

i  4.77  c 

Platinum  

2  1  so 

I     242.8 

Gold  

i  218  8 

Mercury  
Lead  (cast) 

13.60 

850.0 

7OQ   4. 

Silver  

10  ^o 

6s6  i 

Copper  (cast)  
Brass            ... 

8.79 
8  28 

549-4 

C22    8 

Wrought  Iron  

7  68 

480.  o 

Cast  Iron 

721 

4  CO  O 

Steel  

7  84 

400  o 

Tin  (cast) 

7    20 

4C  c  6 

Zinc  (cast)  

6  86 

428  8 

Antimony  

Aluminum 

6.71 
2  so 

419.4 
1^6^ 

WOODS. 


Substance. 

Specific 
Gravity. 

Weight  per 
Cubic  Foot 
in  Pounds. 

Ash  

84C 

C2    80 

Beech  

.8^2 

C7    2C. 

Cedar 

e6i 

T.  C    06 

Cork     

240 

i  c;  oo 

Ebony  (American)  

i.  2,2,1 

82.10 

Lignum-  vitae 

I     3,  27 

82.  70 

Maple  

.7150 

46.88 

Oak  (old) 

I    I  70 

72,    IO 

Spruce  ......                           

.  "JOO 

21.  2*1 

Pine  (yellow) 

660 

41.  2O 

Pine  (white) 

.  t?t\4 

24.6O 

Walnut  

.671 

41.90 

TABLES   AND    FORMULAS. 
LIQUIDS. 


67 


Substance. 

Specific 
Gravity. 

Weight 
per 
Cubic  Foot 
in  Pounds. 

Acetic  Acid 

i  062 

66  4 

Nitric  Acid 

I    217 

76.1 

Sulphuric  Acid  

1.841 

1  15.  i 

Muriatic  Acid 

I    2OO 

75.  0 

Alcohol  

.800 

tJO.O 

Turpentine  

.870 

54-4 

Sea.  \Vater  (ordinary)                  

I.O26 

64.1 

Milk  

1.032 

64.5 

GASES. 

At  32°  F.,  and  under  a  Pressure  of  One  Atmosphere. 


Substance. 

Specific 
Gravity. 

Weight 
per 
Cubic  Foot 
in  Pounds. 

Atmospheric  Air.                      .        .    ... 

I    OOOO 

.  08073 

Carbonic  Acid  

I.  52QO 

.  12344 

Carbonic  Oxide 

0674. 

07810 

Chlorine  

2   4400 

.  19700 

Oxygen 

I    1056 

0802  ^ 

Nitrogen  

07  ^6 

.07860 

Smoke  (bituminous  coal) 

IO2O 

00815 

Smoke  (wood)      .      .  .                      

0900 

.  00727 

*Steam  at  212°  F 

4.7OO 

07700 

Hydrogen 

0602 

ooc  so 

*  The  specific  gravity  of  steam  at  any  temperature  and  pressure  com- 
pared with  air  at  the  same  temperature  and  pressure  is  0.622. 


(IS 


TABLES   AND   FORMULAS. 
MISCELLANEOUS. 


Substance. 

Specific 
Gravity. 

Weight 
per 
Cubic  Foot 
in  Pounds. 

400 

2  ?O 

Glass  (average)                             

2   80 

1  7  c 

Chalk  

2.78 

174 

Granite                                     

2.6? 

166 

Marble                  

2.  70 

169 

Stone  (common)                   

2.  CT2 

1^8 

217 

1  1  1 

Soil  (common)                  

i.  08 

1  24 

Clay                   

1.93 

121 

Brick                                                

I.  QO 

118 

Plaster  Paris  (average)  

2.OO 

12? 

Sand 

I    80 

11^ 

COEFFICIENTS  FOR   FLOW   OF  WATER. 


DISCHARGE  OF  STANDARD   ORIFICES. 


COEFFICIENTS   FOR   CIRCULAR    VERTICAL  ORIFICES. 


Head  h 


Diameter  of  Orifice  in  Feet. 


in  Feet. 

O.O2 

0.04 

0.07 

o.  10 

o.  20 

0.60 

I.  00 

0.4 

0.637 

0.624 

0.618 

0.6 

°-655 

.630 

.618 

.613 

0.601 

Q-593 

0.8 

.648 

.626 

•615 

.610 

.601 

•594 

0.590 

I.O 

•644 

.623 

.612 

.608 

.600 

•595 

•  591 

i-5 

•637 

.618 

.608 

.605 

.600 

•596 

•593 

2.0 

.632 

.614 

.607 

.604 

•599 

•597 

•595 

2-5 

.629 

.612 

.605 

.603 

•599 

.598 

•596 

3-° 

.627 

.611 

.604 

.603 

•599 

•598 

•597 

4.0 

.623 

.609 

•  603 

.602 

•599 

•597 

•596 

6.0 

.6.18 

.607 

.602 

.600 

•598 

•597 

•596 

8.0 

.614 

.605 

.601 

.600 

•598 

.596 

•596 

IO.O 

.611 

.603 

•599 

•598 

•597 

•596 

•595 

20.  o 

.601 

•599 

•597 

•596 

•596 

•596 

•594 

50.0 

•596 

•595 

•594 

•594 

•594 

•594 

•593 

100.  0 

•593 

•592 

•592 

•592 

•592 

•592 

•592 

TABLES  AND   FORMULAS. 


69 


COEFFICIENTS   FOR  SQUARE   VERTICAL   ORIFICES. 


Head  // 
in  Feet. 

Side  of  the  Square  in  Feet. 

O.O2 

0.04 

0.07 

O.  10 

o.  20 

0.60 

I.OO 

0.4 

0.643 

0.628 

0.621 

0.6 

0.660 

.636 

.623 

.617 

0.605 

0.598 

0.8 

.652 

.631 

.620 

.615 

.605 

.600 

°-597 

I.O 

.648 

.628 

.618 

.613 

.605 

.601 

•599 

i-5 

.641 

.622 

.614 

.610 

.605 

.602 

.601 

2.O 

.637 

.619 

.612 

.608 

.605 

.604 

.602 

2-5 

.634 

.617 

.610 

.607 

.605 

.604 

.602 

3-° 

.632 

.616 

.609 

.607 

.605 

.604 

.603 

4.0 

.628 

.614 

.608 

.606 

.605 

.603 

.602 

6.0 

.623 

.612 

.607 

.605 

.604 

.603 

.602 

8.0 

.619 

.610 

.606 

.605 

.604 

.603 

.602 

IO.O 

.6l6 

.608 

.605 

.604 

.603 

.602 

.601 

20.0 

.606 

.604 

.602 

.602 

.602 

.601 

.600 

5O.O 

.6O2 

.601 

.601 

.600 

.600 

•599 

•599 

IOO.O 

•599 

.598 

•598 

•598 

•598 

•598 

•598 

COEFFICIENTS   FOR   RECTANGULAR   ORIFICES 
1    FOOT   WIDE. 


Head  // 
on  Center 
of  Orifice 
in  Feet. 

Depth  of  Orifice  in  Feet. 

o.  125 

0.25 

o.  50 

0.75 

I.OO 

1.50 

2.OO 

0.4 

0.634 

0.633 

0.622 

0.6 

.633 

.633 

.619 

0.614 

0.8 

.633 

•633 

.618 

.612 

0.608 

I.O 

.632 

.632 

.618 

.612 

.606 

0.626 

J-5 

.630 

.631 

.618 

.611 

.605 

.626 

0.628 

2.0 

.629 

.630 

.617 

.611 

.605 

.624 

.630 

2-5 

.628 

.628 

.616 

.611 

.605 

.616 

.627 

3-° 

.627 

.627 

.615 

.610 

.605 

.614 

.619 

4.0 

.624 

.624 

.614 

.609 

.605 

.612 

.6l6 

6.0 

.615 

.615 

.609 

.604 

.602 

.606 

.6lO 

8.0 

.609 

.607 

.603 

.602 

.601 

.602 

.604 

IO.O 

.606 

.603 

.601 

.601 

.601 

.601 

.602 

20.0 

.601 

.601 

.601 

.602 

70 


TABLES   AND    FORMULAS. 


DISCHARGE  OF  WEIRS. 


COEFFICIENTS    FOR    WEIRS    WITH    END    CONTRACTIONS. 


Effective 
Head  in 
Feet. 

Length  of  Weir  in  Feet. 

0.66 

i 

2 

3 

5 

10 

19 

O.I 

0.632 

0.639 

0.646 

0.652 

°-653 

°-655 

0.656 

0.15 

.619 

.625 

.634 

.638 

.640 

.641 

.642 

o.  20 

.611 

.618 

.626 

.630 

.631 

•633 

•634 

0.25 

.605 

.612 

.621 

.624 

.626 

.628 

.629 

0.30 

.601 

.608 

.6l6 

.619 

.621 

.624 

.625 

0.40 

•595 

.601 

.609 

.613 

.615 

.618 

.620 

0.50 

•59° 

-596 

.605 

.608 

.611 

.615 

.617 

0.60 

•587 

•593 

.6oi 

.605 

.608 

.613 

•615 

0.70 

•59° 

•598 

.603 

.606 

.612 

.614 

0.80 

•595 

.600 

.604 

.611 

.613 

0.90 

•592 

•598 

.603 

.609 

.612 

I.OO 

•59° 

•595 

.601 

.608 

.611 

I.  2 

•585 

•591 

•597 

.605 

.610 

1.4 

.580 

•587 

•594 

.602 

.609 

1.6 

.582 

•591 

.600 

.607 

NOTE.—  The  head  given  is  the  effective  head, 
velocity  of  approach  is  small,  h  is  neglected. 


When  the 


COEFFICIENTS  FOR  WEIRS  WITHOUT  END  CONTRACTIONS. 


Effective 


Length  of  Weir  in  Feet. 


neau  irr 
Feet. 

19 

IO 

7 

5 

4 

3 

2 

o.  10 

0.657 

0.658 

0.658 

0.659 

0.15 

•643 

.644 

.645 

•645 

0.647 

0.649 

0.652 

o.  20 

.635 

.637 

•637 

.638 

.641 

.642 

•645 

0.25 

.630 

.632 

.633 

•634 

.636 

.638 

.64I 

0.30 

.626 

.628 

.629 

.631 

•633 

.636 

•639 

0.40 

.621 

.623 

•625 

.628 

.630 

.633 

.636 

0.50 

.619 

.621 

.624 

.627 

.630 

•633 

.637 

0.60 

.618 

.620 

.623 

.627 

.630 

•634 

.638 

0.70 

.618 

.620 

.624 

.628 

.631 

•635 

.640 

0.80 

.618 

.621 

.625 

.629 

•633 

.637 

•643 

0.90 

.619 

.622 

.627 

•631 

.635 

•639 

.645 

I.OO 

.619 

.624 

.628 

•633 

.637 

.641 

.648 

I.  2 

.620 

.626 

.632 

.636 

.641 

.646 

1.4 

.622 

.629 

•634 

.640 

.644 

1.6 

.623 

.631 

.637 

.642 

.647 

NOTE.— The  head  given  is  the  effective  head,  //+-<//.     When  the 
velocity  of  approach  is  small,  h  may  be  neglected. 


TABLES   AND    FORMULAS. 


71 


ffi 


5 


o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o 

OO-t-MCNwi-iO    oco  co"  t^o"  m  -t^co  0   M   M 

o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o 

ini^coor^mmcoOcot^-or^OcoOOOOO 

oooooooooooooooooooo 
o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o  o 
o^ooooooooooooooooooooo 

I 

1 

COC^C^dCIWC^MWC'IC^C^MMMh-IMMI-IMI-IM 

j  OOOOOOOOOOOOOOOOOOOOOO 

o 

lr>Mco'o>u->in^f-t-cON   M   £?  O   C?cc  r^o  in  -r  co  co  M 

o*  c?  o  o  o  o  o  q  q  q  q  q  q  q  q  q  q  q  q  q  q  q 

q  q  q  o  o  o  o  o  q  q  o  o  q  o  q  o  o  q  q  q  q  q 

?  P^  CO  CO  g"^    g"g  'gg'g'g'gggg    2"  "2    M"M    w1  M" 

mOQmOOOinOmincocoOcoOOOOOOO 
O   ^00   in4?i   Ocoo  m-tcoco<N   w   5    Oco  r--O  vn 

o>ooo1o>o'oio>ooqqqqqqooqqqq 

O   >n  co       co'o          CO       O    CO  Q    co  O  O 

S          qqqi-ii-iwMcoinocoqco«noqin__ 

|     .,     II  II  II  II  II  II  II  II  II  II  II  II  II  II  H  II  II  II  II  II  H  H 
8    ^^'-^^---^coocjocoo^oo^co^^ 


7-2 


TABLES   AND    FORMULAS. 


COEFFICIENTS    FOR    ANGULAR    BENDS. 

a"  =  angle  of  bend  in  degrees. 


a°  = 

10° 
.017 

20° 
.046 

40° 
•139 

60° 
•364 

80° 
•74 

90° 
.984 

1  00° 
1.26 

I  IO° 

1.56 

T2O° 

1.86 

130° 
2.16 

140° 
2-43 

*5°° 

C   — 

2.81 

COEFFICIENTS    FOR    CIRCULAR    BENDS. 

r  =  radius  of  pipe.     R  =  radius  of  bend. 


r 
-R- 

.  i 

.  2 

•3 

•4 

•5 

.6 

•7 

.8 

•9 

i  .0 

c'  = 

•I31 

.138 

•158 

.206 

.294 

.440 

.661 

•977 

i  .  408 

1.978 

COEFFICIENTS    FOR    DARCY'S    FORMULA. 


Diameter 
of  Pipe 
in  Inches. 

Coefficients  for 
Rough  Pipes. 

Coefficients  for 
Smooth  Pipes. 

3 

0.00080 

0.00040 

4 

.00076 

.00038 

6 

.00072 

.00036 

8 

.00068 

.00034 

10 

.00066 

.00033 

12 

.00066 

.00033 

14 

.00065 

.00033 

16 

.00064 

.00032 

24 

.00064 

.00032 

3° 

.00063 

.00032 

36 

.00062 

.00031 

48 

.00062 

.00031 

TABLES   AND    FORMULAS. 


7:5 


THE   PROPERTIES  OF  SATURATED 
STEAM. 


c 

'S 

Quantities  of  Heat  in  British 
Thermal  Units. 

"o 

Volume. 

3  "~J 

,q 

I-S 

V  &  n  " 

"£3 

°N 

S 

S^  d£ 

rt  S3 

^ 

"c3  "*"*  O 

** 

CO 

o  § 

S 

S  ^  £  '"^ 

«  o- 

£  S 

ti^? 

S  . 

<D 
> 

13  o 

OQ  w 

£•%£  g 

If 

II 

Sfis 

•!->  O 

ts  £ 

£ 

03 

*fl 

u 

5*S| 

II 

s, 

s 

0} 

!|| 

'£  C  -4-» 

9 

S 
HI 
"3 

'S 

1 

Pi 

£~ 

g^1" 

i 

I 

^ 

0 

P^  c5"o 

I 

2 

3 

4 

5 

6 

7 

8 

/ 

t 

» 

L 

H 

IP 

F 

* 

I 

IO2.OI8 

70.040 

1043.015 

"I3-055 

.003027 

330.4 

20623 

2 

I26.3O2 

94.368 

1026.094 

1120.462 

.005818 

171.9 

10730 

3 

141.654 

109.764 

1015.380 

1125.144 

.008522 

H7-3 

7325 

4 

153.122 

121.  271 

1007.370 

1128.641 

.011172 

89.51 

5588 

5 

162.370 

I30-563 

1000.899 

1131.462 

.013781 

72.56 

453° 

6 

170.173 

138.401 

995-441 

1133.842 

.016357 

61.14 

3816 

7 

176.945 

I45-2I3 

990.695 

1135.908  .018908 

52.89 

33°2 

8 

182.952 

151.253 

986.485 

1137.740.021436 

46.65 

2912 

9 

188.357 

156.  699 

982.690 

1  139.389 

.023944 

41-77 

2607 

10 

193.284 

l6l.66o 

979.232 

1140.892 

.026437 

37.83 

2361 

1  1 

197.814 

166.  225 

976.050 

1142.275 

.02891  i 

34-59 

2159 

12 

2O2.OI2 

170.457 

973.098 

"43-555 

.031376 

31-87 

1990 

13 

205.929 

174.402 

970.346 

1144.748 

.033828 

29-56 

1845 

14 

209.604 

178.112 

967-757 

1145.869 

.036265 

27-58 

1721 

14.69 

2I2.OOO 

180.531 

966.069 

1  146.600 

.037928 

26.37 

1646 

15 

213.067 

181.608 

965.318 

1  146.926 

.038688 

25-85 

1614 

16 

216.347 

184.919 

963.007 

1  147.926 

.041109 

24-33 

15*9 

17 

219.452 

188.056 

960.818 

1148.874 

•0435T9 

22.98 

1434 

1  8 

222.424 

191.058 

958.721 

1149.779 

.045920 

21.78 

1359 

19 

225.255 

193.918 

956.725 

1150.643 

.048312 

20.70 

1292 

TABLES    AND    FORMULAS. 


I 

2 

3 

4 

5 

6 

7 

8 

p 

/ 

9 

L 

// 

w 

V 

R 

20 

227.964 

196.655 

954.814 

1  151.469 

.050696 

19-73 

1231.0 

22 

233.069 

201.817 

951.209 

1153.026 

•055446 

18.04 

1126.0 

24 

237.803 

206.610 

947.861 

1154.471 

.060171 

16.62 

1038.0 

26 

242.225 

21  1.089 

944-73° 

1155.819 

.064870 

15-42 

962.3 

28 

246.376 

215.293 

941.791 

1157.084 

•069545 

14.38 

897.6 

3° 

250.293 

219.261 

939.019 

1158.280 

.074201 

13.48 

841.3 

32 

254.002 

223.021 

936.389 

1159.410 

.078839 

12.68 

791.8 

34 

257.523 

226.594 

933-891 

1  160.485 

.083461 

11.98 

748.0 

36 

260.883 

23O.OOI 

931.508 

1  161.509 

.088067 

11.36 

708.8 

33 

264.093 

233-26I 

929.227 

1162.488 

.092657 

10.79 

673-7 

40 

267.168 

236.386 

927.040 

1  163.426 

.097231 

10.28 

642.0 

42 

270.  122 

239-389 

924.940 

1164.329 

.101794 

9.826 

6i3-3 

44 

272.965 

242.275 

922.919 

1165.194 

.106345 

9-403 

587.0 

46 

275.704 

245.061 

920.968 

1166.029 

.110884 

9.018 

563-0 

48 

278.348 

247.752 

919.084 

1166.836 

.115411 

8.665 

540-9 

5° 

280.904 

250-355 

917.260 

1  167.615 

.119927 

8.338 

520.5 

52 

283.381 

252.875 

9*5-494 

1168.369 

•124433 

8.037 

5OI-7 

54 

285.781 

255-32I 

913.781 

1169.102 

.128928 

7-756 

484.2 

56 

288.III 

257-695 

912.118 

1169.813 

•133414 

7.496 

467.9 

58 

290.374 

26O.OO2 

910.501 

1170.503 

.137892 

7.252 

452.7 

60 

292-575 

262.248 

908.928 

1171.  176 

.142362 

7.024 

438.5 

62 

294.717 

264.433 

907.396 

1171.829 

.146824 

6.811 

425-2 

64 

296.805 

266.566 

905.900 

1  172.466 

.151277 

6.610 

412.6 

66 

298.842 

268.644 

904.443 

1173.087 

•155721 

6.422 

400.8 

68 

300.831 

270.674 

903.020 

1173.694 

.  160157 

6.244 

389.8 

70 

302.774 

272.657 

901.629 

1174.286 

.164584 

6.076 

379-3 

72 

304.669 

274-597 

900.  269 

1174.866 

.169003 

5-91? 

369-4 

74 

306.526 

276.493 

898.938 

H75-431 

•173417 

5-767 

360.0 

76 

308.344 

278.350 

897-635 

1175-985 

.177825 

5.624 

351-1 

78 

310.123 

280.'  170 

896.359 

1176.529 

.  182229 

5.488 

342.6 

80 

311.866 

281.952 

895.108 

1177.060 

.186627 

5-358 

334-5 

82 

3I3-576 

283.  701 

893.879 

1177.580 

.  191017 

5-235 

326.8 

84 

3I5-25° 

285.414 

892.677 

1  1.78.091 

.195401 

5.118 

3r9-5 

86 

316.893 

287.096 

891.496 

1178.592 

.199781 

5.006 

3I2-5 

88 

318.510 

288.750 

890-335 

1179.085 

•204155 

4.898 

305-8 

TABLES   AND    FORMULAS. 


I 

2 

3 

4 

5 

6 

7 

8 

p 

t 

f 

L 

H 

W 

V 

R 

90 

320.094 

290-373 

889.196 

1179.569 

.  208525 

4.796 

299-4 

•  92 

32I-653 

291.970 

888.075 

1  180.045 

.212892 

4.697 

293.2 

94 

323-  183 

293-539 

886.972 

1  180.511 

.217253 

4.603 

287.3 

96 

324.688 

295.083 

885.887 

1180.970 

.221604 

4.513 

281.7 

98 

326.  169 

296.601 

884.821 

1  181.422 

•225950 

4.426 

276.3 

100 

327.625 

298.093 

883.773 

1181.866 

-230293 

4-342 

271.1 

I05 

331.169 

301.731 

881.214 

1  182.945 

.241139 

4-147 

258.9 

I  IO 

334-582 

305.242 

878.744 

1183.986 

.251947 

3-969 

247.8 

1*5 

337-874 

308.621 

876.371 

1184.992 

.262732 

3.806 

237-6 

120 

341.058 

311.885 

874.076 

1185.961 

.273500 

3-656 

228.3 

125 

344-I36 

3i5-°5* 

871.848 

1186.899 

.284243 

3-518 

219.6 

130 

347.121 

318.121 

869.688 

1187.809 

.294961 

3-390 

21  1.6 

135 

35°-OI5 

321.105 

867.590 

1188.695 

•305659 

3.272 

204.2 

140 

352-827 

324.003 

865.552 

Il89-555 

•316338 

3-  J6i 

T97-3 

!45 

355-562 

326.823 

863.567 

i  190.390 

.326998 

3-058 

190.9 

'5° 

358-223 

329.566 

861.634 

1191.200 

•337643 

2.962 

184.9 

1  60 

363-346 

334-850 

857.912 

1192.762 

.358886 

.786 

J73-9 

170 

368.226 

339-892 

854-359 

1194.251 

.380071 

.631 

164.3 

180 

372.886 

344.708 

850.963 

1195.671 

.401201 

•493 

'SS-6 

190 

377-352 

349-329 

847.703 

1197.032 

.422280 

.368 

147.8 

200 

381.636 

353.766 

844.573 

II98-339 

•443310 

.256 

140.8 

2IO 

385.759 

358.041 

841.556 

IJ99-597 

•464295 

.154 

J34-5 

22O 

389-736 

362.168 

838.642 

1200.810 

•485237 

.061 

128.7 

2.30 

393-575 

366.  152 

835.828 

1201.980 

.506139 

.976 

123-3 

240 

397-285 

370.008 

833-  103 

1203.  in 

•527003 

.898 

118.5 

250 

400.883 

373-75° 

830.459 

1204.  209 

.547831 

.825 

1  14.0 

260 

404.370 

377-377 

827.896 

1205.273 

.568626 

-759 

109.8 

270 

407.755 

380.905 

825.401 

1206.306 

•58939° 

•697 

105.9 

280 

41  1.048 

384.337 

822.973 

1207.310 

.610124 

•639 

102.3 

290 

414.250 

387.677 

820.609 

1208.286 

.630829 

•585 

99.0 

300 

4i7-37i 

390-933 

818.305 

1209.  238 

.651506 

-535 

95-8 

TABLES   AND    FORMULAS 


MISCELLANEOUS  TABLES. 


STANDARD    DIMENSIONS    OF    WROUGHT-IRON 
STEAM,    GAS,    AND     WATER     PIPES. 


Nominal 
Diameter 
in  Inches. 

Thickness 
in  Inches. 

Actual 
Internal 
Diameter 
in  Inches. 

Actual 
External 
Diameter 
in  Inches. 

Threads 
per  Inch. 

H 

Pitch  of 
Threads. 

I 

.068 

.270 

•4°5 

27 

•°37 

i 

.088 

•364 

•54° 

18 

.056 

1 

.091 

•494 

•675 

18 

.056 

t 

.109 

.623 

.840 

14 

.071 

1 

•113 

.824 

1.050 

14 

.071 

i 

•134 

1.048 

I.3I5 

"i 

.087 

I| 

.140 

1.380 

i.  660 

"i 

.087 

4 

•145 

1.61  1 

1.900 

»* 

.087 

2 

•154 

2.067 

2-375 

»i 

.087 

«* 

.204 

2.468 

2-875 

8 

.125 

3 

.217 

3.061 

3-5°° 

8 

.125 

3i 

.226 

3-548 

4.000 

8 

•125 

4 

•237 

4.026 

4.500 

8 

•I25 

4l 

.247 

4.508 

5.000 

8 

•'25 

5 

•259 

5-°45 

5-563 

8 

•I25 

6 

.280' 

6.065 

6.625 

8 

.125 

7 

.301 

7.023 

7.625 

8 

.125 

8 

.322 

7.982 

8.625 

8 

-I25 

9 

•344 

9.001 

9.688 

8 

.125 

10 

.366 

10.019 

10.750 

8 

.125 

TABLES  AND  FORMULAS. 


STANDARD    PIPE    FLANGES. 


Inside 
Diam. 
of 
Pipe. 

Thick- 
ness of 
Pipe. 

Diam. 
of 
Bolts. 

Length 
of 
Bolts. 

No.  of 
Bolts. 

Thick- 
ness of 
Flange. 

Diam.   of 
Bolt 

Circle. 

Diam. 
of 
Flange. 

2.O 

.409 

f 

2.O 

4 

f 

4-75 

6.0 

2-5 

.429 

| 

2.25 

4 

H 

5-25 

7.0 

3-° 

.448 

f 

2-5 

4 

t 

6.0 

7-5 

3-5 

.466 

f 

2-5 

4 

H 

6-5 

8-5 

4.0 

.486 

1 

2-75 

4 

H 

7-25 

9.0 

4-5 

.498 

f 

3-° 

8 

H 

7-75 

9-25 

5 

•525 

f 

3-o 

8 

H 

8-5 

IO.O 

6 

-563 

f 

3-° 

8 

i 

9.625 

I  I.O 

7 

.600 

t 

3-25 

8 

*A 

IO-75 

I2-5 

8 

•639 

f 

3-5 

8 

4 

11-75 

13-5 

9 

.678 

1 

3-5 

I  2 

'*  ' 

13.0 

'S-0 

10 

•713 

1 

3-625 

12 

'  ft 

14-25 

16.0 

12 

•79° 

1 

3-75 

12 

'1 

16.5 

19.0 

14 

.864 

i 

4-25 

12 

if 

18.75 

21.0 

J5 

.904 

i 

4-25 

16 

4 

20.  o 

22.25 

16 

.946 

i 

4-25 

16 

'A 

21.25 

23-5 

18 

i.  020 

I* 

4-75 

16 

^ 

22.75 

25.0 

20 

1.090 

*i 

5-° 

20 

'tt 

25.0 

27-5 

22 

1.180 

'i 

5-5 

20 

«H 

27-25 

29-5 

24 

1.250 

l± 

5-5 

20 

«l 

29-5 

32.0 

26 

1.300 

«* 

5-75 

24 

2 

3i-75 

34-25 

28 

1.380 

i± 

6.0 

28 

*A 

34-Q 

36-5 

3° 

1.480 

If 

6.25 

28 

2| 

36.0 

38.75 

36 

1.710 

if 

6-5 

32 

2f 

42.75 

45-75 

42 

1.870 

4 

7-25 

36 

*f 

49-5 

52-75 

48 

2.  170 

4 

7-75 

44 

2f 

56.0 

59-5 

TABLES  AND    FORMULAS. 


SPECIFIC    HEAT    OF    SUBSTANCES. 


Substance. 

Specific 
Heat. 

Substance. 

Specific 
Heat. 

Water 

I    OOOO 

Ice 

Sulphur   
Iron  
Copper 

.  2026 

.1138 

OQ  1  I 

Steam  (superheated) 
Air  
Oxygen 

.4805 
•2375 

2  I  7  ? 

Silver  

.  CK  7O 

Hydrogen  

7  4000 

Tin 

oc62 

Carbon  monoxide 

24.70 

Mercury  

Ot  1  7 

Carbon  dioxide 

2  I  7O 

Lead 

O"?  14. 

Nitrogen 

24.^8 

CONSTANTS    FOR    APPARENT    CUT-OFFS    USED 
IN  DETERMINING  M.  E.  P. 


Cut-off. 

Constant. 

Cut-off. 

Constant. 

Cut-off. 

Constant. 

% 

.566 

3/S 

.771 

2/3 

.917 

Ys 

.603 

•4 

.789 

•7 

.926 

ti 

•659 

/4 

.847 

% 

•937 

•3 

.708 

.6 

•895 

.8 

•944 

/3 

•743 

5/B 

.904 

7A 

•95i 

RIVETED  JOINTS  OF  BOILERS. 


Thick- 
ness of 

Diam- 

Diam- 
eter of 

Pitch. 

Efficiency  of  Joint. 

Plate. 

Rivet. 

Hole. 

d 

Single. 

Double. 

Single, 

Double. 

X* 

$/s" 

U" 

2" 

3" 

.66 

•77 

IT" 

-H" 

y\" 

2iV" 

3/8" 

.64 

.76 

ysff 

y±" 

ii!" 

2^" 

31A" 

.62 

•75 

rV" 

H; 

fa 

2  iV" 

3/s" 

.60 

•74 

1/2  " 

7/8" 

™ 

2^" 

3l/2/> 

•58 

•73 

TABLES  AND    FORMULAS.  79 

POSITIONS  OF  ECCENTRIC  RELATIVE  TO  CRANK. 


Kind  of 

Kind  of 
Rocker- 

Angle  Between 
Crank  and 

Position  of 
Eccentric  Rela- 

Valve. 

Arm. 

Eccentric. 

tive  to  Crank. 

I.  ... 

Direct.  .  . 

Direct  

90°  -j-  angle    of 

advance  

Ahead  of  crank. 

II... 

Direct.  .  . 

Reversing.. 

90°  —  angle    of 

advance  

Behind  crank. 

III.. 

Indirect  . 

Direct  

90°  —  angle    of 

advance  

Behind  crank. 

IV... 

Indirect. 

Reversing.. 

90°  -f-  angle    of 

advance  

Ahead  of  crank. 

DIAMETERS  OF  STEAM    AND    EXHAUST  PIPES. 


Diam.  of  cylinder.  .  . 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

3° 

Diam.  of  steam  pipe  . 

3 

z% 

4 

4^ 

5 

6 

6 

7 

7 

8 

9 

Diam.  of  exhaust  pipe 

3/2 

4 

5 

6 

6 

7 

8 

9 

9 

9 

10 

PISTON  SPEEDS  OF  STEAM  ENGINES. 

Ft.  per  min. 

Small  stationary  engines 300  to  600. 

Large  stationary  engines 600  to  1,000. 

Corliss  engines 400  to  750. 

Locomotives 600  to  1,200. 

RATIO  OF  GRATE  AREA  OF  BOILER   TO 
HORSEPOWER. 

Ratio.         Average. 

Plain  cylindrical 5     to  .  7 

Flue 4    to  .5 

Multitubular 4    to  .6 

Water  tube 3 

Vertical 6    to  .  7 

Locomotive  .  .  .    .01  to  .06 


.6 

•45 

•5 

•3 


80         TABLES  AND  FORMULAS. 

RATIO  OF  HEATING  SURFACE  TO  GRATE  AREA. 

Plain  cylindrical 1 2  to     15 

Flue 20  to    25 

Multitubular 25  to    35 

Vertical 25  to    30 

Water  tube 35  to    40 

Locomotive 50  to  100 

RATIO  OF  HEATING  SURFACE  TO  HORSEPOWER. 

Plain  cylindrical 6  to  10 

Flue 8  to  12 

Multitubular 14  to  18 

Vertical 15  to  20 

Water  tube 10  to  12 

Locomotive.  .  i  to    2 


TABLES   AND    FORMULAS. 


81 


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VI 

TABLES   AND   FORMULAS  83 

RULES  AND   FORMULAS. 


FORMULAS  USED   IN  ALGEBRA. 

Let  a  and  b  be  any  two  quantities,  then, 

(a  +  by  =  «a  +  '*"b  +  £'•  (!•)  Art- 

(«  -  £)•  =  a"  -  lab  +  b\  (2.)  Art.  432. 

(0  +  b}(a  -  b)  =  a1  -  V  (3.)  Art.  432. 

^)  =  (a+b)\  (4.)  '  Art.  455. 

^)  =  («-^)3.  (5.)     Ait.  455. 

^  -b9  =(a  +  b)(a  -  b}.  (6.)  Art.  462. 

Let  ax*  -f-  ^-l'  =  ^  be  any  quadratic  equation  (it  must  be 
borne  in  mind  that  b  and  c  may  be  positive  or  negative) ; 

then, 


THE    TRIGONOMETRIC    FUNCTIONS. 
Art.  754. 

,, .          side  opposite 

Rule   l.—Stne  =.  —. &- . 

hypotenuse 

Rule  2. — Side  opposite  =  hypotenuse  x  sine. 

~    .          side  adjacent 

Rule  3. — Cosine  =  —. ^ . 

hypotenuse 

Rule  4. — Side  adjacent  =  hypotenuse  X  cosine. 


84  TABLES  AND  FORMULAS. 

side  opposite 
Rule  5. —  Tangent  =  —r-. — ^V — 

side  adjacent 

Rule  6. — Side  opposite  =  side  adjacent  x  tangent. 

„  side  adjacent 

Rule  7. — Cotangent  =  — r-; —    — = — . 
side  opposite 

Rule  8. — Side  adjacent  =  side  opposite  X  cotangent 

side  opposite 

Rule  9. — Hypotenuse  = v^ . 

sine 

side  adjacent 

Rule  1O. — Hypotenuse  =  —  —. 

cosine 


RULES    FOR     USING    TABLE     OF     LOGARITHMS 
OF    NUMBERS. 
Arts.    625-636. 

I.  To  find  the  Characteristic. — Fora  number  greater 
than  1  the  characteristic  is  one  less  than   the  number  of  in- 
tegral places  in  the  number.     For  a  number  wtiolly  decimal 
the  characteristic  is  negative,  and  is  numerically  one  greater 
than  the  number  of  ciphers  between  the  decimal  point  and  the 
first  digit  of  the  decimal. 

II.  To  find  the  Logarithm  of  a  Number  not  hav- 
ing more  than  four  figures. — Find  the  first  three  sig- 
nificant figures  of  the  number  whose  logarithm  is  desired  in 
the  left-hand  column  ;  find  the  fourth  figure  in  the  column  at 
the  top  (or  bottom)  of  the  page,  and  in  the  column  under  (or 
above']  this  figure,  and  opposite  the  first  three  figures  previously 
found,  will  be  the  mantissa,  or  decimal  part,  of  the  logarithm. 

The  characteristic  being  found  as  described  above,  write  it  at 
the  left  of  the  mantissa,  and  the  resulting  expression  will  be 
the  logarithm  of  the  required  number. 

III.  To  find  the    Logarithm    of   a    Number    con- 
sisting of  five  or  more  figures. 

(a)  If  the  number  consists  of  more  than  five  figures,  and 
tJie  six tli  figure  is  5  or  greater,  increase  t/ie  fifth  figure  by  1, 
and  write  ciphers  in  place  of  the  sixth  and  remaining  figures. 


TABLES  AND  FORMULAS.  85 

(6)     Find  the  mantissa  corresponding  to  the  logarithm  of 
the  first  four  figures,  and  subtract  tJiis  mantissa  from  the 
next  greater  mantissa  in   the  table ;   the   remainder  is  the 
difference. 

(c)  Find  in  the  secondary  table  headed  P.  P.  a  column 
headed  by  the  same  number  as  tJiat  just  found  for  the  differ- 
ence, and  in  this  column  opposite  tJie  number  corresponding  to 
the  fifth  figure  (or  fifth  figure  increased  by  1)  of  the  given 
number  (this  figure  is  always  situated  at  the  left  of  the 
dividing  line  of  the  column]  will  be  found  the  P.  P.  (propor- 
tional part}  for  that  number.  The  P.  P.  thus  found  is  to  be 
added  to  t/ie  mantissa  found  in  (b},  and  the  result  is  the 
mantissa  of  the  logarithm  of  the  given  number,  as  nearly  as 
may  be  found  with  five-place  tables. 

IV.  To  find  a  Number  whose  Logarithm  is 
given. — 

(a)  Consider  the  mantissa  first.  Glance  along  the 
different  columns  of  the  table  wJiicJi  are  Jieadcd  O  until  the 
first  two  figures  of  the  mantissa  are  found.  TJicn  glance 
doivn  the  same  column  until  the  third  figure  is  found  (or  1 
less  than  the  third  figure}.  Having  found  the  first  three 
figures,  glance  to  the  right  along  the  row  in  which  they  are 
situated  until  the  last  three  figures  of  the  mantissa  are  found. 
Then  the  number  which  heads  the  column  in  ivJiicli  the  last 
three  figures  of  the  mantissa  are  found  is  the  fourth  figure 
of  t/ie  required  number,  and  iJic  first  tJiree  figures  lie  in  the 
column  headed  N,  and  in  the  same  row  in  which  lie  the  last 
three  figures  of  the  mantissa. 

(b}  If  the  mantissa  cannot  be  found  in  the  table,  find  the 
mantissa  wJiicJi  is  nearest  to,  but  less  than,  the  given  mantissa, 
and  which  call  the  next  less  mantissa.  Subtract  the  next  less 
mantissa  from  the  next  greater  mantissa  in  the  table  to  obtain 
the  difference.  Also  subtract  tJie  next  less  mantissa  from  the 
mantissa  of  the  given  logarithm,  and  call  the  remainder  the 
P.  P.  Looking  in  the  secondary  table  headed  P.  P.  for  the 
column  headed  by  the  difference  just  found,  find  the  number 
opposite  the  P.  P.  just  found  (or  the  P.  P.  corresponding  most 


SO  TABLES  AND  FORMULAS. 

nearly  to  that  just  found]  ;  this  number  is  the  fifth  figure  of 
the  required  number  ;  tJie  fourth  figure  will  be  found  at  the 
top  of  t/te  column  containing'  the  next  less  mantissa,  and  the 
first  three  figures  in  the  column  headed  X,  and  in  the  same 
row  which  contains  the  next  less  mantissa. 

(c)  Having  found  the  figures  of  the  number  as  above 
directed,  locate  the  decimal  point  by  tlie  rules  for  the  c/iarac- 
t eristic,  annexing  ciphers  to  bring  the  number  up  to  the  re- 
quired number  of  figures  if  t  lie  e/iaracteristic  is  greater  than  4- 


RULES    FOR    USING    TRIGONOMETRIC    TABLES. 

Given,  an  angle,  to  find  its  sine,  cosine,  tangent, 
and  cotangent. 

Rule  1 1. — Find  in  the  table  the  sine,  cosine,  tangent,  or  co- 
tangent corresponding  to  the  degrees  and  minutes  of  the  angle. 

For  the  seconds,  find  the  difference  of  the  values  of  the  sine, 
cosine,  tangent,  or  cotangent  taken  from  the  table,  between 
whicli  the  seconds  of  the  angle  fall ;  multiply  this  difference  by 
a  fraction  whose  numerator  is  the  number  of  seconds  in  the 
given  angle,  and  whose  denominator  is  60. 

If  sine  or  tangent,  add  this  correction  to  the  value  first  found; 
if  cosine  or  cotangent,  subtract  the  correction.  Art.  756. 

Given,  the  sine,  cosine,  tangent,  or  cotangent  to 
find  the  angle  corresponding. 

To  find  the  angle  corresponding  to  a  given  sine,  cosine, 
tangent,  or  cotangent  whose  exact  value  is  not  contained  in 
the  table : 

Rule  12. — Find  the  difference  of  the  two  numbers  in  the 
table  between  whicli  the  given  sine,  cosine,  tangent,  or  co- 
tangent falls,  and  use  the  number  of  parts  in  this  difference 
as  the  denominator  of  a  fraction. 

Find  the  difference  between  the  number  belonging  to  the 
smaller  angle,  and  the  given  sine,  cosine,  tangent,  or  cotangent, 
and  use  the  number  of  parts  in  the  difference  jiist  found  as  the 
numerator  of  the  fraction  mentioned  above.  Multiply  this 
fraction  by  60,  and  the  result  will  be  the  mimbcr  of  seconds  to 
be  added  to  the  smaller  angle.  Art.  758. 


TABLES  AND  FORMULAS.         87 
RULES  FOR  MENSURATION. 


THE   TRIANGLE. 

Rule. —  TJie    area    of   any    triangle   eqttals   one-half  t/ie 
producr  of  the  base  and  the  altitude.     Art.  766. 


THE   QUADRILATERAL. 

To  find  the  area  of  a  parallelogram: 

Rule. —  The  area  of  any  parallelogram  equals  the  product 
of  the  base  and  the  altitude.  Art.  111. 

To  find  the  area  of  a  trapezoid : 

Rule. —  The  area  of  a  trapezoid  equals  one-half  the  sum  of 
the  parallel  sides  multiplied  by  the  altitude.  Art.  778. 

To  find  the  area  of  an  irregular  figure  bounded  by  straight 
lines: 

Rule. — Divide  the  figure  into  triangles,  and  find  the  area 
of  cac/i  triangle  separately.  T/ie  sum  of  the  areas  of  all  the 
triangles  u>ill  be  the  area  of  the  figure.  Art.  779. 


THE   CIRCLE. 

To  find  the  circumference  or  diameter  of  a  circle: 

Rule. —  The  circumference  of  a  circle  equals  the  diameter 
multiplied  by  3. 1416.  Art.  78O. 

Rule. —  The  diameter  of  a  circle  equals  the  circumference 
divided  by  3.1416.  Art.  78O. 

To  find  the  length  of  an  arc  of  a  circle: 

Rule. —  The  length  of  an  arc  of  a  circle  equals  the  circum- 
ference of  the  circle  of  which  the  are  is  apart  multiplied  by  the 
number  of  degrees  in  the  arc,  and  divided  by  360.  Art.  781 . 

To  find  the  area  of  a  circle: 

Rule. — Square  the  diameter,  and  multiply  by .  7854-  Art. 
782. 

Given,  the  area  of  a  circle  to  find  its  diameter: 

Rule. — Divide  the  area  by  .  7854,  and  extract  the  square 
root  of  the  quotient.  Art.  783. 


88  TABLES  AND  FORMULAS. 

To  find  the  area  of  a  sector : 

Rule. — Divide  the  number  of  degrees  in  the  arc  of  a  sector 
by  360.  Multiply  the  result  by  the  area  of  the  circle  of  which 
the  sector  is  a  part.  Art.  784. 

To  find  the  area  of  a  segment  of  a  circle : 

Rule. — Draiv  radii  from  the  center  of  the  circle  to  the 
extremities  of  the  arc  of 'the  segment ;  find  the  area  of  the 
sector  thus  formed,  subtract  from  this  the  area  of  the- triangle 
formed  by  the  radii  and  the  chord  of  the  arc  of  the  segment, 
and  the  result  is  the  area  of  the  segment.  Art.  785. 


THE    ELLIPSE. 

To  find  the  perimeter  of  an  ellipse: 

Rule. — Multiply  the  major  axis  by  1.83,  and  t lie  minor  axis 
by  1.315.  The  sum  of  the  results  will  be  the  perimeter.  Art. 
788. 

To  find  the  exact  area  of  an  ellipse: 

Rule. —  The  area  of  an  ellipse  is  equal  to  the  product  of  its 
two  diameters  multiplied  by  .7854-  Art.  789. 

To  find  the  area  of  any  plane  figure : 

Rule. —  The  area  of  any  plane  figtire  may  be  found  by 
dividing  it  into  triangles,  quadrilaterals,  circles  or  parts  of 
circles,  and  ellipses,  finding  the  area  of  each  part  separately, 
and  adding  them  together.  Art.  79O. 


THE   PRISM   AND   CYLINDER. 

To  find  the  area  of  the  convex  surface  of  any  right  prism 
or  right  cylinder: 

Rule. — Multiply  the  perimeter  of  the  base  by  the  altitude. 
Art.  8O3. 

To  find  the  volume  of  a  right  prism  or  cylinder: 

Rule. —  The  volume  of  any  right  prism  or  cylinder  equals 
the  area  of  the  base  multiplied  by  the  altitude.     Art.  8O4. 


TABLES  AND  FORMULAS.  89 

THE   PYRAMID   AND   CONE. 

To  find  the  area  of  a  right  pyramid  or  right  cone: 
Rule. —  The  convex  area  of  a  rigJit  pyrainid  or  cone  equals 

the  perimeter  of  the  base  multiplied  by  one-half  the  slant 

height.     Art.  8O9. 

To  find  the  volume  of  a  right  pyramid  or  cone: 

Rule. —  The  volume  of  a  rigJit  pyramid  or  cone  equals  2 'he area 

of  the  base  multiplied  by  one-third  of  the  altitude.     Art.  81O. 


THE   FRUSTUM   OF   A   PYRAMID   OR   CONE. 

To  find  the  convex  area  of  a  frustum  of  a  right  pyramid 
or  right  cone: 

Rule. —  The  convex  area  of  a  frustum  of  a  right  pyramid 
or  right  cone  equals  one -half  the  sum  of  the  perimeters  of  its 
bases  multiplied  by  the  slant  height  of  the  frustum.  Art. 
814. 

To  find  the  volume  of  the  frustum  of  a  pyramid  or  cone: 

Rule. — Add  the  areas  of  the  upper  base,  the  lower  base,  and 
the  square  root  of  the  product  of  the  areas  of  the  two  bases  ; 
multiply  this  sum  by  one-third  of  the  altitude.  Art.  815. 


THE  SPHERE. 

To  find  the  area  of  the  surface  of  a  sphere : 

Rule. —  The  area  oftJie  surface  of  a  sphere  equals  the  square 
of  the  diameter  multiplied  by  3. 1416.     Art.  817. 

To  find  the  volume  of  a  sphere: 

Rule. —  The  volume  of  a  sphere  equals  the  cube  of  the  diam 
eter  multiplied  by  .5236.     Art.  818. 


FORMULAS    USED    IN    ELEMENTARY 
MECHANICS. 


UNIFORM   MOTION. 

Let  5  =  the  length  of  space  passed  over  uniformly ; 

/  =  the  time  occupied  in  passing  over  the  space  S; 
V  =  the  velocity. 


90  TABLES  AND  FORMULAS. 

V=  — .  (7.)     Art.  859. 

5=  Vt.          (8.)     Art.  859. 
t  =  y.  (9.)     Art.  859. 


MASS,  WEIGHT,   AND   GRAVITY. 

If  the  mass  of  the  body  be  represented  by  ///,  its  weight 
by  W,  and  the  force  of  gravity  at  the  place  where  the  body 
was  weighed  by  g,  we  have 

weight  of  body  W 

mass  =  -. -. rf-,  or  m  =  — .  (1O.)     Art.  888. 

force  of  gravity  g 


FORMULAS  FOR   GRAVITY   PROBLEMS. 

Let  W=  weight  of  body  at  the  surface; 

w  =  weight  of  a  body  at  a  given  distance  above  or 

below  the  surface ; 
d  —  distance  between  the  center  of  the  earth  and  the 

center  of  the  body ; 
R  =  radius  of  the  earth  =  4,000  miles. 

Formula  for  weight  when  the  body  is  below  the  surface: 

wR=dW.          (11.)     Art.  891. 

Formula  for  weight  when  the  body  is  above  the  surface: 
wd*=WR\  (12.)     Art.  891. 


FALLING   BODIES. 

Let  g  =  force  of  gravity  =  constant  accelerating  force  due 

to  the  attraction  of  the  earth ; 
/  =  number  of  seconds  the  body  falls ; 
v  =  velocity  at  the  end  of  the  time  t; 
JL  —  distance  that  a  body  falls  during  the  time  t. 

v-gt.  (13.)     Art.  896. 

That  is,  the  velocity  acquired  by  a  freely  falling  body  at  the 
end  of  t  seconds  equals  32. 16  multiplied  by  the  time  in  seconds 

f  =  ~.  (14.)     Art.  896. 


TABLES  AND  FORMULAS.  (J1 

That  is,  the  number  of  seconds  during  which  a  body  must 
have  fallen  to  acquire  a  given  velocity  equals  the  given  velocity 
in  feet  per  second  divided  by  32.  16. 

h  =  ^-         (15.)     Art.  896. 

£> 

That  is,  the  height  from  which  a  body  must  fall  to  acquire 
a  given  velocity  equals  the  square  of  the  given  velocity  divided 
by  2x32.16. 

v-^^gh.  (16.)     Art.  896. 

That  is,  the  velocity  that  a  body  will  acquire  in  falling 
through  a  given  height  equals  the  square  root  of  the  product 
of  twice  32.  16  and  the  given  height. 


(17.)     Art.  896. 

That  is,  the  distance  a  body  will  fall  in  a  given  time  equals 
32.16  -r-  2  multiplied  by  the  square  of  the  number  of  seconds. 


t  =  \.  (18.)     Art.  896. 

<5> 

That  is,  the  time  it  will  take  a  body  to  fall  through  a  given 
height  equals  the  square  root  of  twice  the  height  divided  by 
32.  16. 


CENTRIFUGAL    FORCE. 

The  value  of  the  centrifugal  force  of  any  revolving  body, 
expressed  in  pounds,  is 

F  =  .00034  W  R  N*;          (19.)     Art.  9O3. 

in  which  F  =  centrifugal  force ; 

W  =.  total  weight  of  body  in  pounds; 

R  =  radius,  usually  taken  as  the  distance  be- 
tween the  center  of  motion  and  the  cen- 
ter of  gravity  of  the  revolving  body,  in 
feet; 

N  =  number  of  revolutions  per  minute. 


92  TABLES  AND  FORMULAS. 

THE   CENTER   OF  GRAVITY   OF  TWO   BODIES. 

Let    /  =  the  distance  between  the  centers  of  the  bodies; 
/,  =  the  short  arm  ; 
w  =  weight  of  small  body  ; 
W=  weight  of  large  body. 


THE  EFFICIENCY   OF   A   MACHINE. 

Let  F  =  the  force  applied  to  the  machine  ; 
V  =  the  velocity  ratio  of  the  machine  ; 
W=  the  weight  actually  lifted  or  equivalent  resist- 

ance overcome; 
E  =  the  efficiency  of  the  machine  ; 

W 
Then,  E  =  -.  (22.)     Art.  95O. 


WORK. 

If  the  force  necessary  to  overcome  the  resistance  be  repre- 
sented by  F,  the  space  through  which  the  resistance  acts  by 
vS,  and  the  work  done  by  [7,  then  U  —  F  S. 

\iW—  the  weight  of  a  body,  and  //  =  the  height  through 
which  it  is  raised,  U  =  W  h.  Hence  the  work  done 

U=FS=Wh.  (23.)     Art.    953. 


POWER. 

The  power  of  a  machine  may  always  be  determined  by 
dividing  the  work  done  in  foot-pounds  by  the  time  in  minutes 
required  to  do  the  work;  i.  e. , 

Power  =  ^-.  (24.)     Art.  954. 


KINETIC   ENERGY. 

Let  W=  the  weight  of  the  body  in  pounds; 
v  —  its  velocity  in  feet  per  second ; 
h  =  the  height  in  feet  through  which  the  body  must 
fall  to  produce  the  velocity  v; 

W 
m  —  the  mass  of  the  body  =  — .     (See  formula  1O.) 


TABLES  AND   FORMULAS.  93 

The  work  necessary  to  raise  a  body  through  a  height  h  is 
Wh.     The  velocity  produced  in  falling  a  height  //  is 

v  —  4/2  gh,  and  //  =  — .      (See  formulas  15  and  16.) 

*  v*  w 

Therefore,  work  =  W  h  =  W —  =  \  x  —  X  v1  =  ±mv* 

%g  g 

or  ]Vh  =  %jnv*.  (25.)     Art.  957. 


DENSITY. 

The  density  of  a  body  is  its  mass  divided  by  its  volume 
in  cubic  feet. 

Let  D  be  the  density;  then  the  density  of  a  body  is 

m  W  W 

D  =  ^.     Since  m  =  _,/?=  ~^~  (26.)     Art.  962. 


RULES  AND  FORMULAS  USED  IN  HYDRAULICS. 


PASCAL'S    LAW. 

Rule. — -The  pressure  per  unit  of  area  exerted  anywhere 
upon  a  mass  of  liquid  is  transmitted  undiminished  in  all 
directions,  and  acts  with  the  same  force  upon  all  surfaces  in 
a  direction  at  right  angles  to  those  surfaces.  Art.  968. 


THE  GENERAL  LAW  FOR  THE  DOWNWARD  PRESSURE 
UPON  THE  BOTTOM  OF  ANY  VESSEL. 

Rule. —  The  pressure  upon  the  bottom  of  a  I'csscl  containing 
a  fluid  is  independent  of  the  shape  of  the  -vessel,  and  is  equal 
to  tJie  weight  of  a  prism  of  the  fluid  whose  base  has  the  same 
area  as  the  bottom  of  the  vessel,  and  whose  altitude  is  the 
distance  between  the  bottom  and  the  upper  surface  of  the  fluid 
pins  t/ie  pressure  per  unit  of  area  upon  the  upper  surface  of 
the  fluid,  multiplied  by  the  area  of  the  bottom  of  the  vessel. 
Art.  971.  

GENERAL   LAW   FOR    UPWARD   PRESSURE. 

Rule. —  The  upward  pressure  on  any  submerged  horizontal 
surface  equals  the  weight  of  a  prism  of  the  liquid  whose 
base  has  an  area  equal  to  the  area  of  the  submerged  surface, 


94  TABLES   AND    FORMULAS. 

and  whose  altitude  is  the  distance  between  the  submerged 
surface  and  the  upper  surface  of  the  liquid  plus  the  pressure 
per  unit  of  area  on  the  upper  surface  of  the  fluid,  multiplied 
by  the  area  of  the  submerged  surface.  Art.  973. 

GENERAL  LAW  FOR  LATERAL  PRESSURE. 

Rule. —  The  pressure  upon  any  vertical  surface  due  to  tJie 
weight  of  a  liquid  is  equal  to  the  weight  of  a  prism  of  the 
liquid  whose  base  has  the  same  area  as  the  vertical  surface, 
and  whose  altitude  is  the  depth  of  the  center  of  gravity  of 
the  vertical  surface  below  the  level  of  the  liquid. 

Any  additional  pressure  is  to  be  added,  as  in  the  previous 
cases.  Art.  975. 

GENERAL    LAW    FOR    PRESSURE. 

Rule. —  The  pressure  exerted  by  a  fluid  in  any  direction 
upon  any  surface  is  equal  to  the  weight  of  a  prism  of  the 
fluid  whose  base  is  tJie  projection  of  the  surface  at  rig/it 
angles  to  the  direction  considered,  and  whose  height  is  the 
depth  of  the  center  of  gravity  of  the  surface  below  the  level 
of  the  liqiiid.  Art.  979. 


SPECIFIC    GRAVITY. 

Let  IV  be  the  weight  of  the  solid  in  air  and  W  the  weight 
in  water;  then,  W  —  W  =  the  weight  of  a  volume  of  water 
equal  to  the  volume  of  the  solid,  and 

W 
Sp.  Gr.  =  j^rjy,-  (27.)     Art.  982. 


If  the  body  be  lighter  than  water,  a  piece  of  iron  or  other 
heavy  substance  must  be  attached  to  it  sufficiently  heavy  to 
sink  both.  Then  zvcigh  both  bodies  in  air  and  both  in  water. 

Let  W  =  weight  of  both  bodies  in  air; 
W  =  weight  of  both  bodies  in  water; 
w  =  weight  of  light  body  in  air; 
W^  =  weight  of  heavy  body  in  air; 
Wt  =  weight  of  heavy  body  in  water. 


TABLES  AND    FORMULAS.  95 

Then,  the  specific  gravity  of  the  light  body  is  given  by 

sP-Gr-  =  (^-?r)-(n/-^)-        (27"-)   Art-983- 

To  find  the  specific  gravity  of  a  liquid: 

Weigli  an  empty  flask ;  fill  it  with  water,  then  weigh  it, 
and  find  the  difference  between  the  two  results ;  this  will 
equal  the  weight  of  the  water.  Then  weigh  the  flask  filled 
ivit/i  the  liquid,  and  subtract  t/ic  weight  of  tJic  flask ;  the 
result  is  the  weight  of  a  volume  of  the  liquid  equal  to  the 
volume  of  the  water.  The  weight  of  the  liquid  divided  by 
the  weight  of  the  water  is  the  specific  gravity  of  the  liqiiid. 

Let  W  =  the  weight  of  the  flask  and  liquid; 
IT'  =  the  weight  of  the  flask  and  water; 
iv  =  the  weight  of  the  flask. 

Then,  Sp.  Gr.  =  ^— ^-  (276.)     Art.  984. 


FORMULAS    FOR    FLOW    OF    WATER. 


MEAN    VELOCITY. 

Let  Q  =  the  quantity  in  cubic  feet  which  passes  any  sec- 
tion in  1  second; 

A  =  the  area  of  the  section  in  square  feet ; 

vm  =  the  mean  velocity  in  feet  per  second. 
Then,  Q  =  Avm,     (28*.)     Art.  989. 

and  v»—%-  (28*.)     Art.  989. 


VELOCITY    OF   EFFLUX  FROM   AX   ORIFICE. 

Let   v  —  the  velocity  of  efflux  in  feet  per  second; 
h  =  the  head  in  feet  on  the  orifice  considered; 
//,  —  the  head  equivalent  to  a  pressure/; 
IV  =  the    weight    of    the    water    in    pounds    flowing 
through  the  aperture  per  second. 

ixr  „.! 

The  kinetic  energy  of  the  issuing  water  =  — . 


96  TABLES   AND    FORMULAS. 

The  work  the  issuing  water  can  do  =  W  h. 

Wh  =  -^-,  or  v  =  ^Tg~h. 
*•  '  S 

//,  =  -frrr,   where  hl  is  in  feet,   and  /  in  pounds  per 
.  4o4: 

square  inch. 

h  =  •—-=,   where  h    is  in  feet,   and  p  in  pounds  per 
b*.  O 

square  foot. 
h-^-h^  —  the  total  head. 

v  =  tf*g(hl  +  7J).  (29.)     Art.  991. 

If  a  is  the  area  of  a  large  orifice  in  the  bottom  of  a  small 
vessel  whose  area  is  A,  the  velocity  is 


Art.  993. 


THEORETICAL   RANGE    OF    A    JET. 

Let  //  =  head  on  center  of  orifice ; 

y  =  vertical  height  of  orifice  above  the  surface  where 

the  water  strikes; 
R  =  range. 

Then,         R  =  tfUTy.  (3O.)     Art.  992. 


FLOW    THROUGH   ORIFICES. 

Velocity  of  the  Jet. 

Let  v  =  theoretical  velocity ; 

v'  —  actual  maximum  velocity ; 

c'  =  coefficient  of  velocity ; 

h  =  head  on  center  of  orifice ; 

g  —  acceleration  due  to  gravity  —  32.16. 

'if  =  c'  v  =  c'  \fogh.  (32.)     Art.  994. 

An  average  value  of  c'  is  .98. 


TABLES  AND   FORMULAS.  97 

Discharge  of  an  Orifice. 

Let  Q  —  theoretical  discharge; 
Q'  =  actual  discharge  ; 
a  —  area  of  orifice  ; 
c"  =  coefficient  of  discharge  ; 
//  =  head  on  center  of  orifice; 
g  =  acceleration  dtie  to  gravity  —  32.16. 

An  average  value  of  c"  is  .61.     Then, 
Q  =  c"  Q  =  c"a  4/2^7:  =  .  61  a  ^Tgli.  (33.)     Art.  994. 

Discharge  of  Standard  Orifices. 

Let  Q  —  discharge  in  cubic  feet  per  second; 

d   =  diameter  of  a   circular  or  length    of  a   side  of 

a  square  orifice  in  feet  ; 
d'  =  depth  of  a  rectangular  orifice  in  feet; 
/;    =  breacjtfi  of  a  rectangular  orifice  in  feet  ; 
Ji   =  head  on  the  center  of  a  circular  or  of  a  square 

orifice  in  feet; 
//a  =  head  on  the  upper  edge  of  a  rectangular  orifice 

in  feet; 
//a  =  head  on  the  lower  edge  of  a  rectangular  orifice 

in  feet; 
c   =  coefficient  of  discharge  (see  tables  of  Coefficients 

of  Discharge  for  Standard  Orifices)  ; 
g   =  acceleration  due  to  gravity  =  32.16. 

For  a  circular  vertical  orifice, 

Q  =  .  7854  d*c  4/2^7;=  6.299  d*  c  \fJi.         (340.)    Art.  996. 
For  a  square  vertical  orifice, 

Q  =  cd*  yTp  =  8.02  c  dn-  \Hi.  (340.  )     Art.  997. 

For  a  rectangular  vertical  orifice, 


(34^.)     Art.  998. 
If  the  head  h  on   the    center  of  a   rectangular   vertical 


98  TABLES  AND  FORMULAS. 

orifice  is  greater  than  4  //,  the  discharge  may  be  computed 
by  the  formula 

Q  =  c  b d^gli  =  8.Cr>fl><t 4//7.  (34</. )     Art.  998. 

For  approximate  computations  the  value  of  c  that  may  be 
used  in  formulas  34r  and  34^/  is  c  —  .015. 
Discharge  of  a  Submerged  Rectangular  Orifice. 

Let  Q  =  discharge  in  cubic  feet  per  second ; 
b  =  breadth  of  orifice  in  feet ; 
d  =  depth  of  orifice  in  feet ; 
ho  —  the  difference  in  the  level  of  the  water  on  the 

two  sides  of  the  orifice  in  feet ; 
g   —  acceleration  due  to  gravity  =  33.10. 
Then, 
Q  =  .  015  b  d4/~Zgh0  =  4. 932  b  d  \/k0.         (34r. )     Art.  999. 


THE   DISCHARGE   OF    WEIRS. 

Let  /   =  length  of  the  weir  in  feet; 
H  —  measured  head  in  feet'; 
v  =  velocity  with   which   the   water  approaches  the 

weir  in  feet  per  second; 
It  =  head  equivalent  to  the  velocity  with  which  the 

water  approaches  the  weir  in  feet; 
c  =  coefficient  of  discharge  (see  tables  of  Coefficients 

of  Discharge  for  Weirs)  ; 

Q  =  theoretical  discharge  in  cubic  feet  per  second  ; 
Q  =  actual  discharge  in  cubic  feet  per  second. 
The  theoretical  discharge  per  second  is 

Q  =  %\frgl(H+/tf.  (35*.)     Art.  1006. 

If  there  is  no  velocity  of  approach,  this  becomes 

Q  =  \\/^glHl.  (35*.)     Art.  1OO6. 

The  actual  discharge  for  weirs  with  end  contractions  is 
given  by  the  formulas 


(36*.  )     Art.  1OO6. 
and 

Art.  1OO6. 


TABLES    AND    FORMULAS.  09 

For  weirs  without  end  contractions,  the  formulas  are 
Q>  =  c%\i*Tgl(H+  1.4//)§  =  5.347  <:/(#  +  1.4//)*, 

(37  a.)     Art.  1OO6. 

and     0'=rf4/2]r///*=  5.347  <:///*.     (376.)    Art.  1OO6. 

The  velocity  of  approach  is  the  mean  velocity  with 

which  the   water   flows   through    the    canal  leading  to   the 

weir.      If  A  is  the  area  of  the  cross-section  of  the  water  in 

this  canal,  we  have  v  =  ^-,  from  which  we  see  that  Q'  must 

be  determined  approximately  by  assuming  i>  —  0,  and  then 
use  this  value  of  Q'  to  find  i<.  V  may  also  be  measured 
approximately  by  means  of  a  float  on  the  water  in  the  canal 
or  stream. 

v"1 
Having  found  v,  we  have  the  equivalent  head  //  =  -—  = 

.  01555  if.  (See  Arts.  99O  and  991.)  Since  v  is  small  with 
a  properly  constructed  weir,  it  is  usually  neglected,  unless 
great  accuracy  is  required. 


FLOW    OF    WATER    THROUGH    PIPES. 

Let    /    =  length  of  pipe  in  feet ; 

d   —  diameter  of  pipe  in  feet; 

dl  —  diameter  of  pipe  in  inches ; 

i'  =  mean  velocity  of  flow  through  pipe  in  feet  per 
second ; 

Ji    =  total  head  on  outlet  end  of  pipe  in  feet ; 

//'  —  head  in  feet  equivalent  to  the  velocity  v; 

/;'"=head  in  feet  equivalent  to  the  loss  of  pressure 
at  entrance  to  pipe; 

//IV=head  in  feet  equivalent  to  the  loss  in  pressure 
produced  by  friction  in  pipe ; 

//v  =  head  in  feet  equivalent  to  loss  in  pressure  pro- 
duced by  angular  bends  in  pipe; 

//VI  =  head  in  feet  equivalent  to  loss  in  pressure  pro- 
duced by  circular  bends  in  pipe; 

f  =  a  coefficient  for  loss  of  head  due  to  friction  (see 
table  of  Coefficients  f  for  Smooth  Iron  Pipes); 

;;/   =  a  coefficient  for  loss  of  head  at  entrance ; 

n   =  number  of  bends  in  pipe; 


100  TABLES   AND    FORMULAS. 

c    =  a    coefficient   for   loss    of   head   due    to    angular 

bends    (see    table    of    Coefficients   for   Angular 

Bends); 
c1  —  a  coefficient    for  loss    of  head    due   to   circular 

bends    (see  table    of    Coefficients   for    Circular 

Bends) ; 
Q  =  quantity  discharged  by  pipe  in  cubic  feet  per 

second; 
Q'  =  quantity    discharged    by    pipe    in    gallons    per 

second ; 

r    =  radius  of  pipe  in  feet ; 
R  =  radius  of  circular  bend  in  pipe  in  feet; 
a°  =  number  of  degrees  of  angular  bend  in  pipe. 

General  Formulas. 

Loss  of  head  at  entrance, 

h'"  =  m/i"  =  m^-.  (39.)     Art.  1O2O. 

Loss  of  head  due  to  friction, 

W.-fL.  (400.)     Art.  1021. 


Loss  of  head  due  to  angular  bends, 

hv  =  c^  (4O0.)     Art.  1O23. 

*g 

Loss  of  head  due  to  circular  bends, 

h^  —  c'^-.  (4O^.)     Art.  1O23. 

Total  head, 

/fr  It'          |      /^          r  I      *^  1      ^^ 

~+f  j£-  +  m^r+  nc'^T'  (41rt-)     Art.  1O24. 

Velocity  of  flow, 

c=    /         *s* 


d 
8.02      /—  (42.)     Art.  1O24. 


TABLES    AND    FORMULAS.  101 

If  in  —  .  5  and  there  are  no  sharp  bends, 


/      *gk    .  =  8.02      / ^—     ;     (43.)     Art.  1024e 


d 
and,  when  the  diameter  is  in  inches, 

/         7T77 

Art.  1O25. 


Velocity  Through  Long  Pipes. 

When  the  diameter  is  in  feet, 

v  =  8.02  \'j~.  (44.)     Art.  1O25. 

When  the  diameter  is  in  inches, 

v  =  2.  315  y  '-jfj-.  (44*.)     Art.  1O25. 

Head  Required   to   Produce  a  Given  Velocity. 

General  formula, 


(45.)     Art.  1026. 

When  the  influence  of  bends  is  neglected  and  m  has  the 
value  .5,  the  formula  is 

•  °333  T;2-  45^-       Art- 


When  the  diameter  is  given  in  inches, 

//  =  /l-^L-f.  .0233  T'2.  (45^.)     Art.  1O26. 

O.  oD  it 

The  Quantity  Discharged  from  Pipes. 

When  the  diameter  is  given  in  feet,  the  discharge  in  cubic 
feet  per  second  is 

0=.  7854^7*.  (46.)     Art.  1O27. 

Since  one  cubic  foot  contains  7.48  gallons,  if  the  diameter 
is  in  feet,  we  have 


102  TABLES   AND    FORMULAS. 

Q'  —  .  7854  d*  v  X  7.  48  gallons  per  second  ;  (46«.  ) 

Art.  1O27. 

and  for  the  diameter  in  inches, 
£'  =  .0408  d?  v  gallons  per  second.          (46£.)     Art.  1O27. 

The    Diameter    of    Pipes. 

With  //,  /,  and  d  in  feet  and  the  quantity  Q  in  cubic  feet 
per  second,  the  formula  for  the  diameter  of  a  pipe  without 
sharp  bends  is 


d-  0.479     (1.5^  +  //)~  (47.)     Art.  1O28. 

In  using  this  formula,  take  the  approximate  value  of  f  as 
.02,  and  compute  an  approximate  value  for  c/,  neglecting  the 
term  1.5  d  in  the  second  member  of  the  formula.  With 
this  value  of  d,  find  the  value  of  v  from  the  formula 

v  =  and  find  the  corresponding  value  of  /"from  the 

.  7854  d 

table  of  Coefficients  for  Pipes. 

Repeat  the  computation  for  d  by  placing  the  approximate 
values  of  d  and  f  just  found  in  the  second  member  of  the 
formula.  One  or  two  repetitions  of  this  process  will  give  a 
near  approximation  of  d  from  which  to  select  the  pipe  from 
the  standard  market  sizes. 

For  pipes  whose  length  is  more  than  4,000  times  their 
diameter,  the  following  formula  may  be  used  : 

(4:7  a.)     Art.  1O28. 


FLOW   OF   WATER   IN   CONDUITS   AND   CHANNELS. 

Let  5  =  slope  of  a  conduit  or  channel ; 
h  —  a  given  fall ; 

/  =  distance  in  which  the  fall  1i  occurs ; 
p  =  wetted  perimeter; 
a  —  area  of  water  cross-section ; 
r  =  hydraulic  radius; 
v  =  mean  velocity  of  flow ; 
Q  —  quantity  discharged; 


TABLEvS   AND    FORMULAS.  103 

c  —  a  coefficient  to  be  determined  by  Kutter's  for- 
mula; 

n  —  coefficient  of  roughness  to  be  used  in  Kutter's 
formula  (see  table  of  Coefficients  of  Roughness). 

Formula  for  slope, 

(48.)     Art.  1032. 
Hydraulic  radius, 

(49.)     Art.  1O32. 

Discharge, 

Q  =  ar.  Art.  1O32. 

Mean  velocity, 

v  =  *4&&  (50.)     Art.  1033. 

To  find  the  value  of  c  use  Kutter's  formula, 

03   |    *    |    -00155 

..  •  (51-)     Art.  1033. 


The  value  of  n  to  be  used  in  this  formula  is  to  be  taken 
from  the  following  table  to  correspond  with  the  character 
of  the  channel : 

VALUES   OF   THE  COEFFICIENT   OF   ROUGHNESS. 

For  Use  in  Kutter's  Formula. 
Character  of  Channel.  Value  of  n. 

Clean,  well-planed  timber 009 

Clean,  smooth,  glazed  iron  and  stoneware  pipes oio 

Masonry  smoothly  plastered  with  cement on 

Clean,  smooth  cast-iron  pipe on 

Ordinary  cast-iron  pipe 012 

Unplaned  timber 012 

Selected  sewer  pipes,  well  laid  and  thoroughly  flushed.   .012 

Rough  iron  pipes 013 

Ordinary  sewer  pipes  laid  under  usual  conditions 013 

Dressed  masonry  and  well-laid  brickwork 015 


104  TABLES   AND    FORMULAS. 

Character  of  Channel.  Value  of  n. 

Good  rubble  masonry  and  ordinary  rough  or  fouled 

brickwork 017 

Coarse  rubble  masonry 020 

Gravel,  compact  and  firm 020 

Earth  canals,  well  made  and  in  good  alinement 0225 

Rivers  and  canals  in  moderately  good  order  and  per- 
fectly free  from  stones  and  weeds 025 

Rivers  and  canals  in  rather  bad  condition  and  some- 
what obstructed  by  stones  and  weeds 030 

Rivers  and  canals  in  bad  condition,  overgrown  with 
vegetation  and  strewn  with  stones  and  other 
detritus,  according  to  condition 035  to  .050 


FORMULAS    USED    IN    PNEUMATICS. 


PRESSURE,    VOLUME,     DENSITY,    AND     WEIGHT     OF    AIR 
WHEN   THE  TEMPERATURE  IS  CONSTANT  : 

Mariotte's  Law. —  The  temperature  remaining  the  same, 
the  volume  of  a  given  quantity  of  gas  varies  inversely  as  the 
pressure. 

Let/   =  pressure  for  one  position  of  the  piston; 

/t  =  pressure  for  any  other  position  of  the  piston ; 
v  =  volume  corresponding  to  the  pressure/; 
vl  =  volume  corresponding  to  the  pressure/,. 

Then,  /  v=p,  vv     (53.)     Art.  1O49. 
Let  D  be  the  density  corresponding  to  the  pressure  /  and 
volume   v,    and    Z>,    be    the    density    corresponding  to   the 
pressure  /,  and  volume  i\ ;  then, 

/  :D  =A  =  A»  or/ A=A  A       (54.)     Art.  1O52. 
and  v:Dl  =  ^ :  Z>,    or  v  D  =  ^  Z\.       (55.)     Art.  1O52. 

Thus,  let  IV  be  the  weight  of  a  cubic  foot  of  air  or  other  gas, 
whose  volume  is  v,  and  pressure  is/;  let  W^  be  the  weight 
of  a  cubic  foot  when  the  volume  is  vlt  and  pressure  is  /,; 
then, 

/  W^  =/,  W.      (56.)     Art.  1052. 

v  W  =  v,  W,.     (57.)     Art.  1O52. 


TABLES  AND  FORMULAS.  103 

PRESSURE     A1VD    VOLUME    OF     A    GAS    "WITH    VARIABLE 
TEMPERATURE  : 

Gay-L/ussac's  Law.  —  If  tJie  pressure  remains  constant, 
every  increase  of  temperature  of  1°  F.  produces  in  a  given 
quantity  of  gas  an  expansion  of  ^\-%  of  its  volume  at  32°  F. 

If  the  pressure  remains  constant  it  will  also  be  found  that 
every  decrease  of  temperature  of  1°  F.  will  cause  a  decrease 
of  j^  of  the  volume  at  32°  F. 

Let  v  =  original  volume  of  gas  ; 
i\  =  final  volume  of  gas; 

/  —  temperature  corresponding  to  volume  v; 
/j  =  temperature  corresponding  to  volume  v^ 


Then>         v*  =  v-      (58-}  Art-1054- 


That  is,  the  volume  of  gas  after  heating  (or  cooling)  equals 
the  original  volume  multiplied  by  4.60  plus  the  final  tempera- 
ture divided  by  460  plus  the  original  temperature. 

=  the  original  tension; 
=  the  corresponding  temperature; 
l  —  final  tension  ; 
/j  =  final  temperature. 


Then,  A=/-  (59.)     Art.  1O55. 


Let    /  =  pressure  in  pounds  per  square  inch  ; 
V=  volume  of  air  in  cubic  feet; 
T  =  absolute  temperature  ; 
W  =  weight  in  pounds. 

Then,  /  V=  .37052  T.     (6O.)     Art.  1O56. 

If  the  weight  of  the  air  be  greater  or  less  than  1  pound, 
the  following  formula  must  be  used  : 

/  F=.  37052  W  T.     (61.)     Art.  1O57. 
Let  /,,    f7,,   and    7",  represent  the  pressure,  volume,   and 
temperature  of  the  same   weight  of  air  in  another  state; 
then, 

-  =  ^-.         (62.)     Art.  1O58. 


100  TABLES   AND    FORMULAS. 

MIXTURE    OF    TWO     GASES    HAVING    UNEQUAL    VOLUMES 
AND    PRESSURES. 

Let  i'  and  /  be  the  volume  and  pressure,  respectively,  of 
one  of  the  gases. 

Let  ?',  and  p^  be  the  volume  and  pressure,  respectively, 
of  the  other  gas. 

Let  V  and  P  be  the  volume  and  pressure,  respectively,  of 
the  mixture.  Then,  if  the  temperature  remains  the  same, 

VP=vp  +  vlpl.  (63.)     Art  1O62. 


MIXTURE    OF   TWO    VOLUMES    OF   AIR    HAVING    UNEQUAL 
PRESSURES,    VOLUMES,    AND    TEMPERATURES. 

If  a  body  of  air  having  a  temperature  /,,  a  pressure/,,  and 
a  volume  i\  be  mixed  with  another  volume  of  air  having  a 
temperature  t^  a  pressure  /2,  and  a  volume  7'2,  to  form  a 
volume  V  having  a  pressure  Pt  and  a  temperature  /,  then, 
either  the  new  temperature  /,  the  new  volume  V,  or  the  new 
pressure  P  may  be  found,  if  the  other  two  quantities  are 
known,  by  the  following  formula,  in  which  T^  T^  and  T 


(64.)  Art.  1063. 


FORMULAS  USED  IN  STRENGTH  OF 
MATERIALS. 


UNIT    STRESS,    UNIT    STRAIN,    AND    COEFFICIENT    OF 
ELASTICITY. 

Let  P—  the  total  stress  in  pounds; 

A  =  area,  of  cross-section  in  square  inches; 
5  =  unit  stress  in  pounds  per  square  inch ; 

/  =  length  of  body  in  inches ; 

^  =  elongation  in  inches; 

s  =  unit  strain ; 

R  =  coefficient  of  elasticity. 

5  =      ,  or  P=  A  S.  (65.)     Art.  1 1O3. 


TABLES   AND    FORMULAS.  107 


s=,ore=ts.  (66.)     Art.  11O4. 


j, 


Art.  1110. 


STRBIVGTH   OF   PIPES  AND   CYLINDERS. 

Let  d  —  inside  diameter  of  pipe  in  inches; 
/  =  length  of  pipe  in  inches; 
p  —  pressure  in  pounds  per  square  inch; 
P  =  total  pressure;  then,  P  =  p  Id; 
t  —  thickness  of  pipe; 
5  =  working  strength  of  the  material. 
For  longitudinal  rupture 
//</=2//S,  or 

/  d=  a  /  S.  (68.)     Art.  1 123. 

For  transverse  rupture 

pd=±tS.  (69.)     Art.  1124. 

Since,  for  longitudinal  rupture,  p  d=  2  /  5,  it  is  seen  that 
a  cylinder  is  twice  as  strong  against  transverse  rupture  as 
against  longitudinal  rupture. 

For  pipes  and  cylinders  whose  thickness  is  greater  than 
-1*5-  of  the  radius,  use  the  following  formula,  in  which 
r  =  the  inner  radius,  and  the  other  letters  have  the  same 
meaning  as  before. 

/  =  -^-f  (70.)     Art.  1 1 25. 

The  following  formula  gives  the  collapsing  pressure  in  Ib 
per  sq.  in.  for  wrought-iron  pipe: 

/  =9,  GOO,  000  *-.  (71.)     Art.  1126. 


MOMENT  OF  INERTIA,  RESISTING  MOMENT,  AND  BENDING 
MOMENT   OF  BEAMS. 

Let  /  —  moment  of  inertia; 
A  —  area  of  cross-section ; 
r  =  radius  of  gyration; 


108  TABLES   AND    FORMULAS. 

c  =  distance  from  neutral  axis  to  outermost  fiber; 
54  =  ultimate  strength  of  flexure; 
f  =  factor  of  safety ; 
M  —  bending  moment. 

I=Ar\  (72.)     Art.  1154. 

The  resisting  moment  is  given  by  the  expression 

S    .  S    .         „/ 

—  A  r  =  —  /,  or  5  — • 
c  c  c 

For  the  bending  moment 

M-$J-c,  (73.)     Art.  1156. 

and,  when  a  factor  of  safety  is  used, 

M=^.  (74.)     Art.  1159. 


DEFLECTION   OF   A   BEAM. 

Let  a  =  a  constant  depending  on  the  manner  of  loading 

the  beam  and  the  condition  of  the  ends; 
s  =  the  deflection; 
E  =  the  coefficient  of  elasticity; 
/  =  the  length  of  the  beam  in  inches; 
W •=•  the  total  weight  supported  in  pounds; 
/  =  moment  of  inertia  about  the  neutral  axis. 

W  /* 
s  =  a.  (75.)     Art.  1162. 


STRENGTH   OF  COLUMNS. 

Let    W=  load  on  a  column; 

52  =  ultimate  strength  for  compression; 

A  —  area  of  section  of  column ; 

f  —  factor  of  safety ; 

/  =  length  of  column  in  inches; 

g  =  a  constant  to  be  taken  from  table; 

/  =  least  moment  of  inertia  of  cross-section; 

b  —  length  of  longer  side  of  a  rectangular  column; 


TABLES   AND    FORMULAS.  100 

d  =  length  of  shorter  side  of  a  rectangular  column, 

or  the  diameter  of  a  circular  column; 
c  =  length  of  one  side  of  a  square  column. 


Art.  1169. 


For  a  circular  column 

/ 


*  1/-3l83^/i   ,/•  3183  H7V.  3183  Hy      16^1 
S,  ^        \     A  £•    / 

(78.)     Art.  1171. 
For  a  rectangular  column,  assume  d,  then, 


(79.)     Art.  1171. 


d 


STRENGTH   OF   SHAFTS. 

Let  d  =  diameter  of  a  round  shaft,   or  side  of  a  square 

shaft,  in  inches; 

c  =  a  constant  (see  table  of  Constants  for  Shafting) ; 
ct  =  a  constant  (see  table  of  Constants  for  Shafting) ; 
P—  a  force  applied  at  the  end  of  a  lever  arm  in 

pounds; 

r  —  length  of  lever  arm  in  inches; 
H  —  horsepower  transmitted  by  shaft; 
N  =  number  of  revolutions  per  minute; 
k  —  a  constant  (see  table  of  Constants  for  Shafting) ; 
£,  =  a  constant  (see  table  of  Constants  for  Shafting) ; 
q  =  a  constant   (see  table  of  Constants  for  Shafting) ; 
gl  —  a  constant  (see  table  of  Constants  for  Shafting). 

For  all  solid  shafts  below  11  inches  in  diameter  use  the 
formula 

c^  (80.)     Art.  1173. 


110 


TABLES   AND    FORMULAS. 


If  the  diameter  of  a  wrought-iron  shaft  is  greater  than 
12.4",  of  a  cast-iron  shaft  greater  than  10.3",  or  of  a  steel 
shaft  greater  than  13.6",  use  the  following  formula: 


kl\-.  (81.)     Art.  1174. 

For  a  hollow  (round)  shaft  use  formula  82  or  83. 
p  =  q  /^_pA4\  (82.)     Art.  1  1  74. 

or        H=glN^*~d*\  (83.)     Art.  1174. 

CONSTANTS    FOR    SHAFTING. 


VALUES  OF  c  AND  c,  TO  BE  USED  IN  FORMULA  8O. 


Material 

c 

Round. 

Square. 

Round. 

Square. 

Wrought  Iron  .    
Cast  Iron 

.310 

tC  7 

.272 
•  ^OQ 

4.92 

C.  CO 

4-31 
4.89 

Steel  

.297 

.  260 

4.70 

4.11 

VALUES  OF  k,  Jt,,  q,  AND  q,  TO  BE  USED  IN  FORMULAS 

81,  82,  AND  83. 


Material. 

k 

*i 

9 

9i 

Wrought  Iron  
Cast  Iron     

.0909 
.  1  14"> 

3.62 
4."?6 

i,335 
669 

.0212 
.0106 

Steel 

0828 

1   30 

i  767 

0280 

STRENGTH  OF  ROPES  AND  CHAINS. 

Let  P=  working  or  safe  load  in  pounds; 
C  —  circumference  of  rope  in  inches; 
d=  diameter  of  the  link  of  a  chain  in  inches. 


TABLES   AND    FORMULAS.  Ill 

For  manila  ropes,  hemp  ropes,  or  tarred  hemp  ropes, 

P=100C\  (84.)     Art.  1175. 

For  iron  wire  rope  of  7  strands,  19  wires  to  the  strand, 

P=GOOC\  (85.)     Art.  1176. 

For  the  best  steel  wire  rope,  7  strands,  19  wires  to  the 
strand, 

/>=  1,000  f.  (86.)     Art.  1176. 

For  open-link  chains  made  from  a  good  quality  of  wrought 
iron, 

P=  12,000  d\  (87.)     Art.  1179. 

and  for  stud-link  chains, 

P=  18,000  d*.  (88.)     Art.  1179. 


FORMULAS     USED     IN     SURVEYING. 


RADIUS  OF  A  CURVE. 

To  find  the  radius,  the  degree  being  given: 
Let  R  —  the  length  of  the  required  radius; 

D  =  the  deflection  angle  equal  to  one-half  the  degree 
of  the  given  curve. 

Kf) 

(89'}     Ar 


LENGTH   OF   SUB-CHORDS. 

For  curves  of  short  radii  : 

Let  C  —  the  length  of  the  required  chord; 

R  =  the  radius  of  the  given  curve; 

D  —  the  deflection  angle  of  the  given  curve,  equal  to 
one-half  its  degree. 

C=2^sinZ>.  (9O.)     Art.  125O. 


LENGTH    OF   THE  TANGENT   OF   A   CURVE. 

When  the  radius  and  intersection  angle  are  given: 
Let  T  =  the  length  of  the  required  tangent; 

R  =  the  radius  of  the  given  curve; 

/  =  the  intersection  angle  of  the  given  curve. 

r=A'tan|7.  (91.)     Art.  1251. 


112  TABLES   AND    FORMULAS. 

CHORD    DEFLECTION. 

When  the  length  of  the  chord  and  the  radius  are  given : 
Let  d  —  the  required  chord  deflection  ; 

c  =  the  length  of  the  chord  of  the  given  curve; 

R  —  the  radius  of  the  given  curve. 

<*=.  (92.)     Art.  1255. 


TANGENT    DEFLECTION. 

When  the  length  of  the  tangent,  or  of  its  corresponding 
chord,  and  the  radius  are  given: 

Let  c  =  the  length  of  the  tangent  or  corresponding  chord  ; 
R  =  the  radius  of  the  given  curve. 

tangent  deflection  —  —^.  (93.)     Art.  1255. 

Or,  find  the  chord  deflection  as  in  the  preceding  formula 
and  divide  it  by  2.  The  quotient  is  the  required  tangent 
deflection. 

STADIA   MEASUREMENTS. 

To  find  the  horizontal  distance  between  two  given  points, 
the  distance  between  them  having  been  read  with  the  stadia 
and  the  vertical  angle  taken  : 

Let  D  =  the  corrected  or  horizontal  distance  ; 

c   =  the  constant  ; 
a  k  =  the  stadia  distance  ; 
n  =  the  vertical  angle. 

D  —  c  cos  n  +  a  k  cos3  «.  (94.)     Art.  13O1. 

To  find  the  difference  of  elevation  between  two  given 
points  in  stadia  work  : 

Let  E  =  the  required  difference  in  elevation; 

c    =  the  constant  ; 
a  k  —  the  stadia  distance  ; 
n  =  the  vertical  angle. 

(95.)     Art.  13O1. 


- 
a 


TABLES   AND    FORMULAS.  113 

BAROMETRICAL   LEVELING. 

To  find  the  difference  of    elevation  between  two  points 
with  the  aneroid  barometer: 

Let  Z  —  the  difference  of  elevation  between  the  two  given 

stations; 
h  =  the  reading  in  inches  of   the  barometer  at  the 

lower  station; 
//"=the   reading  in  inches  of   the  barometer  at    the 

higher  station; 
/  and  /'  =  the  temperature  (F.  )  of  the  air  at  the  two  stations. 


Z=  (log  //  -  log//)  X  00,384.3  (l 


+ 


(96.)     Art.  1304. 


RULES    AND    FORMULAS    USED    IN    SURVEYING 
AND    MAPPING. 

Rule  for  Balancing  a  Survey.  —  As   the   sum   of  all 

tJic  courses  is  to  any  separate  course,  so  is  the  whole  difference 
in  latitude  to  the  correction  for  that  course.  A  similar 
proportion  corrects  the  departures.  Art.  1315. 

Rule  for  Double  Longitudes.  —  Tlic  double  longitude 
of  the  first  course  is  equal  to  its  departure. 

The  double  longitude  of  the  second  course  is  equal  to  the 
double  longitude  of  the  first  course  plus  the  departure  of  that 
course  plus  the  departure  of  tJic  second  course. 

The  double  longitude  of  the  third  course  is  equal  to  the 
double  longitude  of  the  second  course  plus  the  departure  of 
that  course  plus  the  departure  of  the  course  itself. 

The  double  longitude  of  any  course  is  equal  to  the  double 
longitude  of  tJie  preceding  course  plus  the  departure  of  that 
course  plus  the  departure  of  tJie  course  itself. 

The  double  -longitude  of  the  last  course  (as  well  as  of  the 
first]  is  equal  to  its  departure.  This  result,  u'Jicn  obtained 
by  the  above  rule,  proves  the  accuracy  of  the  calculation  of  the 
double  longitudes  of  all  the  preceding  courses.  Art.  1319. 


114  TABLES   AND    FORMULAS. 

APPLICATION    OF   DOUBLE   LONGITUDES   TO   FINDING 
AREAS. 

1.  Prepare  ten  columns,  and  in   the  first  three  write  the 
stations,  bearings,  and  distances,  respectively. 

2.  Find  the  latitudes  and  departures  of  each  course  by  the 
Traverse  Table,  placing  them  in  the  four  following  columns. 

3.  Balance  them  by  the  above  rule  for  balancing  a  survey, 
correcting  them  in  red  ink. 

4.  Find  the  double  longitudes  by  the  rule  for  double  longi- 
tudes, with   reference  to   a    meridian   passing  tJirong]i    tJie 
extreme  east  or  west  station,  and  place  them  in  the  eighth 
column. 

5.  Multiply  the  double  longitude  of  each  course  by  the  cor- 
rected latitude  for  that  course,  placing  the  north  products  in 
the  ninth  column  and  the  south  products  in  the  tenth  column. 

6..  Add  the  last  two  columns;  subtract  the  smaller  sum 
from  the  larger,  and  divide  the  difference  by  2.  The  quotient 
will  be  the  area  required.  Art.  1321. 


AREAS   OF   IRREGULAR   FIGURES. 

Trapezoidal  Rule.—  Divide  t  lie  figure  into  any  sufficient 
number  of  equal  parts  by  means  of  vertical  lines  called 
ordi  nates;  add  half  the  sum  of  the  t^vo  end  ordinates  to  the 
sum  of  all  the  other  ordinates  ;  divide  by  the  number  of  spaces 
(that  is,  by  one  less  than  the  number  of  ordinates}  to  obtain 
the  mean  ordinate,  and  multiply  t  lie  quotient  by  the  length  of 
the  section  to  obtain  the  area. 

Simpson's  Rule.  —  Divide  the  length  of  the  figure  into 
any  even  number  of  equal  parts,  at  t/ie  common  distance  D 
apart,  and  draw  ordinates  tJirough  the  points  of  division. 
Add  together  the  length  of  the  first  and  the  last  ordinates 
and  call  the  sum  A  ;  add  together  the  even  ordinates  and  call 
the  stun  B  ;  add  together  the  odd  ordinates,  except  the  first 
and  the  last,  and  call  the  sum  C. 


Then,  area  of  figure  -  x  D      Art    1324. 


TABLES   AND    FORMULAS.  115 

VOLUMES   OF    IRREGULAR  SOLIDS. 

To  find  the  volume  included  between  two  parallel  cross- 
sections  whose  areas  are  known, 

Let  A  =  area  of  one  section  in  square  feet; 

B  =  area  of  the  other  section  in  square  feet; 

C  —  distance  between  the  two  sections  in  feet; 

D  =  required  volume  in  cubic  feet. 
Then,  approximately, 

D  =  ^-^-x  C.  (97.)     Art.  1325. 

The  Prismoidal  Formula. — A  more  accurate  fesult 
than  that  given  by  the  last  formula  is  given  by  the 
prismoidal  formula. 

Let  A  =  area  of  one  section  in  square  feet; 

B  •=.  area  of  the  other  section  in  square  feet ; 

M=  the    area   of    the   average   or   mean    section    in 

square  feet; 

L  =  distance  between  the  sections  in  feet ; 
5  =  the  required  volume  in  cubic  feet. 

).  (98.)     Art.  1326. 


LATITUDES   AND  DEPARTURES. 

To  find  the  latitude  and  departure  of  a  course  by  means 
of  a  table  of  sines  and  cosines, 

Latitude  —  distance  x  cosine  bearing.     (99.)    Art.  1338. 
Departure  =  distance  X  sin  bearing.      (1OO.)    Art.  1338. 


FORMULAS    USED    IN    STEAM    AND    STEAM 
ENGINES. 


SPECIFIC    HEAX. 

W  —  weight  of  body  in  pounds ; 
/   —  temperature  before  heat  is  applied ; 
/,  =  temperature  after  heat  is  applied; 
c   =  specific  heat  of  body; 


116  TABLES   AND    FORMULAS. 

U  =  number  of  B.  T.  U.  required  to  raise  temperature  of 
body  from  /  to  /,. 

U=cW(tl-t}.  (101.)     Art.  1379. 


TEMPERATURE   OF   MIXTURES. 

;>,  iv  ^  wt, .  .  .  .  =  weights  of  the  several  substances,  respect- 
ively; 

<:,  cl,  ct, .  .  .  .    —  specific  heats  of  the  substances,  respect- 
ively; 

£,/,,/,,....    =  temperatures  of  the  substances,  respect- 
ively; 

T  =  final  temperature  of  mixture. 

T_wcf  +  w^cltl-\-wtctt  -{-  .  ...^     Art<  1383, 


Mixture   of   Steam    and  Water. 

W  •=  weight  of  steam  in  pounds; 

w  =  weight  of  water  in  pounds; 

tl  =  temperature  of  steam; 

/    =  temperature  of  water; 

T  =  final  temperature  of  mixture; 

L  =  latent  heat  of  steam  at  the  given  temperature. 


WORK    DO1VE    BY    PISTON. 

/  =  net  pressure  on  piston  in  pounds  per  square  inch 
V  •=•  volume  in  cubic  feet  swept  through  by  piston; 
W=  work  done  by  moving  piston. 

W=  144/  V.  (103.)     Art.  1395. 


REAL    AND    APPARENT    CUT-OFF. 

s  =  apparent  cut-off; 

k  =  real  cut-off; 

i  =  clearance  expressed  as  a  per  cent,  of  the  stroke. 

£=*-±i  (104.)     Art.  1457. 


TABLES   AND    FORMULAS.  117 

HORSEPOWER. 

I.  H.  P.  =  indicated  horsepower  of  engine; 

P=  mean  effective  pressure  in  pounds  per  sq.  in.  ; 

A  =  area  of  piston  in  square  inches; 

L  =  length  of  stroke  in  feet; 

N  •=  number  of  strokes  per  minute. 

PT  A  N 

L  H'  R  =  -         105-     Art' 


MEAN  EFFECTIVE  PRESSURE. 

p  =  gauge  pressure; 

k  —  constant  depending  upon  cut-off  (see  table  of 

Constants  used  in  determining  M.  E.  P.); 
M.  E.  P.  =  mean  effective  pressure. 

M.  E.  P.  =  .9  [£(/>+  14.  7)  -  17].  Art.  1496. 


PISTON    SPEED. 

/  =  length  of  stroke  in  inches; 
R  —  number  of  revolutions  per  minute; 
5  =  piston  speed  in  feet  per  minute. 

/  /? 
S  =  —  .  (1O6.)     Art.  1497. 


MECHANICAL   EFFICIENCY   OF   ENGINE. 

I.  H.  P.  =  indicated  horsepower; 

Friction  H.  P.  =  horsepower  absorbed  in  overcoming  fric- 
tion of  engine; 

Net  H.  P.=  I.  H.  P.  —Friction  H.  P.  =  horsepower  avail- 
able to  perform  useful  work  ; 

E  =  efficiency  of  engine. 


STEAM   CONSUMPTION. 

/  —  distance  between  two  points  on  the  indicator  dia- 
gram, one  on  the  expansion  line,  and  the  other  on 
the  compression  line,  both  being  equally  distant 
from  the  vacuum  line ; 


118  TABLES   AND    FORMULAS. 

L  =  length  of  indicator  diagram ; 

a  =  absolute  pressure  of  steam  at  the  two  points  chosen 

W '=  weight  of  a  cubic  foot  at  pressure  a; 

Q  =  steam  consumption  in  pounds  per  I.  H.  P.  per  hour. 

.          (107.)     Art.  1507. 


THERMAL  EFFICIENCY   OF  ENGINE. 

T^=  absolute  temperature  of  steam  entering  cylinder; 
7"a=  absolute  temperature  of  steam  leaving  cylinder; 
E  =  thermal  efficiency. 

E  =  ^7^.  Art.  1512. 


WATER  REQUIRE!)  BY   CONDENSER. 

/j  =  temperature  of  departing  condensing  water; 

/a  =  temperature  of  entering  condensing  water; 

/,  =  temperature  of  the  condensed  steam  upon   leaving 

the  condenser; 
H=  total  heat  of  vaporization  of  one  pound  of  steam  at  the 

pressure  of  the  exhaust  (see  steam  table,  column  5) ; 
W  =  weight  of  water  required  to  condense  a  pound  of  steam. 

W=H~+3*.  (108.)     Art.  1520. 


RATIO   OF   EXPANSION. 

e  =  ratio  of  expansion  in  high-pressure  cylinder ; 
E  =  total  ratio  of  expansion ; 

v  =  volume  of  cylinder  receiving  steam  from  boiler ; 
V  —  volume    of    cylinder    or    cylinders   exhausting    into 
atmosphere  or  condenser, 

£=~-  (109.)     Art.  1527. 


TABLES   AND    FORMULAS.  119 

FORMULAS    USED    IN    STEAM    BOILERS. 


AIR   REQUIRED   FOR   COMBUSTION    AND    HEAT   OF 
COMBUSTION. 

C  —  percentage  of  carbon  in  a  fuel  expressed  in  parts  of 

a  hundred; 
//  —  percentage  of  hydrogen  in  a  fuel  expressed  in  parts 

of  a  hundred  ; 
A  =  cubic  feet  of  air  required  to  burn  a  pound  of  the  fuel. 

A  =  1.5Z(C+3H).  (110.)     Art.  1546. 

B  =  British  thermal  units  produced  by  the  combustion  of 

the  fuel  ; 
U'=  weight  of  water  that  can  be  evaporated  by  a  pound 

of  the  fuel. 

(111.)     Art.  1547. 


STRENGTH    OF    BOILER    SHELLS. 

P  —  gauge  pressure  of  steam,  pounds  per  square  inch  ; 

d  =  diameter  of  shell  in  inches; 

'/  —  length  of  shell  in  inches; 

/   =  thickness  of  material  ; 

S  —  safe  stress  in  material:  9,000  Ib.  for  wrought  iron;. 

11,000  Ib.  for  steel; 

F  =  total  force  tending  to  rupture  the  shell; 
e  =  efficiency  of  joint  (see  table  of  Riveted  Joints). 
F=Pdl.  (112.)     Art.  1603. 

(113.)     Art.  1604. 


d    ' 


HORSEPOWER    OF    BOILERS. 

W=  pounds  of  water  evaporated  per  hour; 
F  —  factor  of  evaporation  (see  table  of  Factors  of  Evapo- 
ration) ; 
H  =  horsepower  of  boiler. 

W F 
H=~~.  (114.)     Art.  1618. 

O4:.  O 


120  TABLES    AND    FORMULAS. 

THE    SAFETY    VALVE. 

A  =  area  of  opening  in  valve-seat  in  square  inches ; 

p  =  blow-off  pressure  of  valve; 

a  =  power  arm  of  lever  valve;  i.e.,  the  distance  of  valve- 
stem  from  fulcrum ; 

d  =  weight  arm  of  lever  valve;  i.  e.,  the  distance  of 
weight  from  fulcrum ; 

H  —  reading  of  spring  scale,  when  the  lever  and  valve  are 
attached  to  it,  at  the  point  where  the  valve-stem 
joins  the  lever; 

P  =  weight  of  ball  hung  on  end  of  lever; 

W=  weight  required  in  a  dead-weight  valve; 

S  =  pounds  of  steam  generated  per  hour. 

W=pA.     (115.)  Art.  1621. 
/  =  -?".     (116.)  Art.  1621. 

/i 

paA=Pd.     Art.  1623. 
(pA  -H)a  =  Pt 


d=(pA-H}a 


(117.)  Art.  1624. 


DRAFT   PRESSURE   OF  CHIMNEY. 

H  =  height  of  chimney  in  feet; 

Ta  =  absolute  temperature  of  air; 

Tc  =  absolute  temperature  of  escaping  gases ; 

/  =  draft  pressure  in  inches  of  water. 


(119.)     Art.  1662. 


6       7.9V 

77? 


TABLES   AND    FORMULAS.  121 

QUALITV    OF   STEAM  (BARREL   CALORIMETER). 

W  =•  weight  of  cold  water  in  barrel; 

w  —  weight    of    mixture    of    steam    and   water    run   into 

barrel; 
/  =  temperature    of    steam    corresponding    to    observed 

pressure  ; 

ti  =  original  temperature  of  cold  water; 
/2  •=  temperature  of  cold  water  after  steam  is  condensed  ; 
L  —  latent   heat  of    a  pound  of    steam  at   the  observed 

pressure  (see  column  4,  steam  tables)  ; 
x  =  portion  of  weight  w  that  is  dry  steam  ; 

Q  =  quality  of  steam  =  —  . 


FORMULAS  USED   IN   WATER-WHEELS. 


THEORETICAL  ENERGY  OF    A   GIVEN   HEAD  AND   WEIGHT 
OF   WATER. 

Let  h  =  available  head ; 

v  =  velocity  the  water  would  attain  if  it  fell  freely 

through  the  height  It; 
W=  weight  of  water; 

g  =  acceleration  due  to  force  of  gravity  =  32.16; 
K  —  theoretical  energy. 

K=  Wh  =  W-.  (121.)     Art.  1727. 


THEORETICAL   POWER. 

Rule. —  To  find  t/ie  theoretical  horsepozver  that  a  given 
quantity  of 'water  will  furnish,  multiply  the  iveight  of  water 
that  falls  in  one  second  by  the  distance  through  which  it  falls, 
and  divide  this  product  by  550 ;  the  quotient  will  be  the 
theoretical  horsepower. 


122  TABLES   AND    FORMULAS. 

Let  H.  P.  =  theoretical  horsepower; 

Q     —  quantity   of  water   falling  in   cubic    feet  per 

second ; 
H    =  total  available  fall  in  feet. 

H.  P.  =  Q  X  6^  X  H  =  .1136  QH.     (122.)    Art.  173O. 
550 


ENERGY    OF    A    JET. 

Let  K  =  energy  of  the  jet; 

W =  weight  of  water  that  flows  from  the  orifice  or 

nozzle  in  one  second ; 

w  =•  weight  of  a  cubic  foot  of  water  —  62.5  pounds; 
a  =  area  of  the  jet  in  square  feet; 
v  =  velocity    of   flow    from    the    orifice    in    feet    per 

second ; 

c  =  coefficient  of  velocity  for  the  orifice; 
h  =  head  on  the  orifice  in  feet ; 
g  =  acceleration  due  to  gravity  =  32.16. 

K=W^--cWh.  (123.)     Art.  1731. 

W=wav.  (124.)     Art.  1731. 

K=™*^  =  cwavh  (125.)     Art.  1731. 


PRESSURE  DUE  TO  IMPACT  AND  REACTION  OF  A  JET. 

Let  P  —  pressure  produced  by  the  impact; 
R  =  reaction  of  the  jet; 
W '  —  weight  of  water  that  flows  from  the  orifice  or 

nozzle  in  one  second ; 

w=  weight  of  a  cubic  foot  of  water  =  62.5  pounds; 
a  =  area  of  the  jet  in  square  feet; 
v  =  velocity    of   flow    from    the    orifice    in    feet    per 

second ; 

c  =  coefficient  of  velocity  for  the  orifice; 
h  =  head  on  the  orifice  in  feet ; 
g  =  acceleration  due  to  gravity  —  32.16. 


TABLES   AND    FORMULAS.  123 

Pressure    on    a    Vertical    Surface. — When    the    jet 
impinges  on  a  vertical  surface  the  pressure  is 

P=wa-  =  Zcwah  =  W"-.  (126.)     Art.  1732. 

o  o 

Reaction. — The  reaction  of  the  jet  is 
R  =  P^wa—  =  <lcwak=  W-.         (127.)     Art.  1732. 

o  o 

Pressure  Produced  by  Change  of  Direction. 

Let  a°  =  the  angle  between  the  original  direction  of  the 
jet  and  its  direction  after  being  deflected. 

The   pressure   exerted  on  the  deflecting   surface  in  the 
original  direction  of  the  jet  is 

P  =  (1  -  cos  a°)  W-.  (128.)     Art.  1734. 

o 

Pressure   on  a  Hemispherical   Cup. — -When  the  jet 
strikes  into  a  hemispherical  cup  a°=  180°,  and  the  pressure  is 

P  =  (l  -  cos  180°)  W-  =  2  W-\ 
g  g 

that  is,  the  pressure  is  twice  as  great  as  the  pressure  pro- 
duced iu lien  the  jet  struck  a  surface  at  right  angles  to  its 
direction  of  motion.  Art.  1734. 

Effect  When  the   Surface  Is  in  Motion. 

Let  v'=  the  velocity  with  which  the  surface  moves  along 
the  line  of  motion  of  the  jet. 

The  pressure  on  the  surface  is 

P  =  (l  -  cosrt0)  w(l  ~~\Z^^--       (129.)    Art.  1735. 
If  the  surface  is  a  hemispherical  cup,  the  pressure  is 
P=.  0622  ~^(z/ -?/)'.  (130.)     Art.  1735. 

The  theoretical  work  of  a  Jet  impinging  in  a  moving 
hemispherical  cup  is  a  maximum  when  the  velocity  of  the  cup 


124  TABLES   AND    FORMULAS. 

is  one-half  tJie  Telocity  of  t/ie  jet,  and  it  is  equal  to  tJie 
theoretical  work  that  zvould  be  done  by  the  energy  due  to  the 
velocity  of  the  water.  Art.  1  735. 


EFFICIENCY. 

Rule  I. —  To  fijid  the  amount  of  work  or  poivcr  that  can 
be  obtained  from  a  given  fall  of  ivatcr  'when  the  efficiency  of 
the  motor  is  given,  multiply  the  theoretical  work  or  power  by 
the  efficiency  expressed  as  a  decimal  fraction,  and  the  product 
will  give  the  available  work  or  power.  Art.  1 737. 

Rule  II. —  To  Jin d  the  quantity  of  water  required  to  fur- 
nish a  given  amount  of  power  with  a  given  efficiency,  divide 
the  theoretical  quantity  of  water  by  t/te  efficiency;  the  quotient 
will  be  the  quantity  required.  Art.  1 737. 


OVERSHOT   WATER-WHEELS. 

Let  H  =  total  fall  of  water ; 

v  =  velocity  of  circumference  of  wheel ; 

ve  =  velocity  with  which  water  enters  wheel ; 

h  =  head  required  to  produce  velocity  of  entry  ve\ 

D  —  outside  diameter  of  wheel ; 

N  =  number  of  revolutions  of  wheel; 

Z '  =  number  of  buckets; 

b  —  breadth  of  buckets; 

d  =  depth  of  buckets; 

Q  —  quantity  of  water  in  cubic  feet  per  second; 

c  =  clearance  between  wheel  and  trough. 

ve  =l$vto%v.  (131.)     Art.  1 743. 

//=l.l|C  (132.)     Art.  1743. 

D=ff-(A  +  c).          Art.  1 743. 
N=lQ.l^.  (133.)     Art.  1743. 

Z=  IQDto  12  Z>.  (134.)     Art.  1743. 

d—  10  inches  to  15  inches.  (135.)     Art.  1743. 

£=3Uo4.  (136.)     Art.  1743. 


TABLES   AND    FORMULAS.  125 

BREAST    WHEELS. 

The  following  rules  may  be  used  for  the  principal  dimen- 
sions of  a  breast  wheel: 

Velocity  of  circumference  of  wheel  v  =  2  feet  per  second 
to  8  feet  per  second.  Velocity  of  entry  ve  =  \%v  to  Zv. 

Depth  of  floats  d  =  10  inches  to  15  inches.  Pitch  of 
floats  t  =  d. 

Diameter  of  wheel,  about  twice  the  total  head. 

Breadth  of  wheel,  b  =  14--^-  to  2-p,  where  Q  is  in  cubic 
a  v          dv 

feet  per  second,  b  and  d  in  feet,  and  v  in  feet  per  second. 
Art.  1749. 

UNDERSHOT   WHEELS. 

Let  H.  P.  =  horsepower; 

v    =  velocity  of  water  in  race  in  feet  per  second; 
vi  =  velocity  of  circumference  of  wheel  in  feet  per 

second ; 
Q  =  quantity  of    water  flowing  through   race  in 

cubic  feet  per  second ; 
F  —  area  of  the  immersed  portion  of  the  float  of  a 

paddle  wheel  in  an  unconfined  current. 

For  a  wheel  in  a  confined  race, 

H.  P.  =  .00215  (v-v,)  7',  Q.  (137.)     Art.  1754. 

For  a  simple  paddle  wheel  in  an  unconfined  current, 

H.  P.  =  .00282  (v  -  v^  v  i\  F.  (138.)      Art.  1755. 


F»OTVCELET'S   WHEEL. 

Let  H=  total  fall  in  feet; 

Q  =  the  quantity  of  water  in  feet  per  second; 

D  —  the  outside  diameter  of  the  wheel  in  feet; 

d  =  depth  of  floats  in  feet ; 

dl  =  depth  of  water  current  entering  the  wheel  in  feet ; 

ve  =  the  velocity  of  the  water  current  entering  the 

wheel  in  feet  per  second ; 
b  =  breadth  of  the  wheel  and  of  the  sluice  in  feet; 


126 


TABLES   AND    FORMULAS. 


vl  —  velocity  of  circumference  of  wheel  in   feet  per 

second  ; 

R  =  radius  of  curvature  of  floats; 
A  =  angle  A  O  B  (see  Fig.  527,  Art.  1  756)  ; 
u   —  number  of  revolutions  per  minute  of  wheel; 
n   =  number  of  floats  in  wheel. 

H  and    Q  must  be  determined  by  actual  measurement; 
the  other  dimensions  may  then  be  made  as  follows: 


d  = 

dl  —  \  foot  to  1  foot  ; 


a  =20°  to  45 


Art.  1757. 


TURBINES. 

In  the  rules  and  formulas  used  to  determine  the  principal 
dimensions  of  reaction  turbines 

Let  Q   =  the  available  quantity  of  water  in  cubic  feet  per 

second ; 

h    =  the  total  available  head  on  the  wheel  in  feet ; 
ve  =  the  velocity  of  the  flow  from  the  guide  buckets 

in  feet  per  second ; 
i>r  =  the  relative  velocity  of  water  entering  the  wheel 

buckets  in  feet  per  second; 
v    =  the  relative    velocity  of   flow   from    the  wheel 

buckets  in  feet  per  second; 
vf  =  the  absolute  velocity  of  the  water  leaving  the 

wheel  buckets; 

vw  =  the  velocity  of  the  wheel  buckets  at  entrance; 
v'w=  the  velocity  of  the  wheel  buckets  at  discharge; 


TABLES   AND    FORMULAS.  12? 

a    —  the  angle  which  the  direction  of  outflow  from 

the  guides  makes  with  the  radius  in  a  radial- 
flow    turbine    or    with  a    perpendicular  to    the 

direction  of  motion  of  the  wheel  buckets  in  an 

axial-flow  turbine; 
a1   =  the  angle  which  the  relative  direction  of  inflow 

to  the  wheel  makes  with  the  same  lines; 
ay   =  the  angle  which  the  relative  direction  of  flow 

from  the  vanes  makes  with  the  same  lines; 
A    =  the  effective  outflow  area  of  guide  passages  in 

square  feet; 
A!  =  the  effective  inflow  area  of  wheel  passages  in 

square  feet ; 
A^  =  the  effective  outflow  area  of  wheel  passages  in 

square  feet; 
A^  =  sectional  area  of  flow  for  draft  tube  in  square 

feet; 
At  =  effective  outflow  area  of  draft  tube  in  square 

feet; 

N  =  the  number  of  revolutions  per  minute ; 
r    =  the  mean  radius  of  an  axial  turbine  in  feet; 
rl  =  the  radius  of  the  wheel  at  inflow,  in  feet,  for  a 

radial-flow  turbine; 
r2  =  the  radius  of  the  wheel  at  outflow,  in  feet,  for  a 

radial-flow  turbine; 

g    =  acceleration  due  to  force  of  gravity; 
Kl  =  a  coefficient  for  finding  the  radius  r  or  r, ,  from 

the  area  A  ; 

k    =  a  coefficient  for  finding  the  velocity  fe; 
P  =  the  pitch  of  the  guide  buckets; 
P^  =  the  pitch  of  the  wheel  buckets; 
Z   =  the  number  of  guide  buckets; 
Z^  =  the  number  of  wheel  buckets; 
//o  =  the  height  of   guide   buckets    in   an    axial-flow 

turbine; 
h\  =  the  height  of    wheel    buckets  in  an  axial-flow 

turbine; 


128  TABLES   AND    FORMULAS. 

x  =  the  distance  between  the  outflow  ends  of  two 
consecutive  guide  buckets,  measured  perpen- 
dicular to  the  direction  of  flow ; 

xt  =  the  distance  between  the  outflow  ends  of  two 
consecutive  wheel  buckets,  measured  perpen- 
dicular to  the  direction  of  flow ; 

/    =  the  thickness  of  guide  buckets  near  ends; 

/j  =  the  thickness  of  wheel  buckets  near  ends; 

s  —  the  part  of  the  distance  x  that  would  be  covered 
by  the  inflow  end  of  one  wheel  bucket,  to  be 
measured  in  the  same  direction  as  .v; 

e    =  the  width  of  outflow  ends  of  guide  buckets ; 

e^  •=.  the  width  of  inflow  ends  of  wheel  buckets; 

e^  •=.  the  width  of  outflow  ends  of  wheel  buckets. 

General  Relations. — The  usual  proportions  and  values 
to  be  used  in  designing  the  different  types  of  wheel  are  as 
follows : 

(a)  For  axial  turbines  using  a  large  quantity  of  water 
under  a  low  head,  where  —  is  greater  than  16  square  feet, 

A   =  70°  to  66°. 
A^  =  70°  to  66°. 

k  =  1  to  l\. 

P  =  10  inches  to  12  inches. 

^  inch  to  f  inch  for  cast  iron. 

£  inch  to  f  inch  for  wrought  iron. 


(b]  For  axial  turbines  using  a  medium  quantity  of  water 

under  medium  head,   where  —  is  greater  than   2  and  less 

ve 

than  16  square  feet, 

A   =  75°  to  70°.  P=  -f—  to  ^-_. 

^,  =  7^  to  73°.  :3'7°       4'5 

„  "  /  =  r  =  same  as  above. 

At  =  .67. 

k   =  1.25  to  1.5.  h  =  //„  =  T-  to  -f-. 

4        4.  o 


TABLES   AND    FORMULAS.  129 

(r)  For  axial  turbines  using  a  small  quantity  of  water 
under  a  high  head,  where  ^-  is  less  than  2  square  feet, 

—  75    to  73  .  p~  4^  t0  g  inches. 

At  —  77°  to  74°.  /  _  /^  _  same  as  above. 
A",  — .07.  r        r 

k   —  1. 5  to  2.  ^  ~~    •"  iO  t0  3 ' 

(;/)  For  radial  inward-flow  turbines,  where  Q  ranges 
from  2.4  to  275  cubic  feet  per  second,  and  h  is  from  3  feet 

to  80  feet, 

A    =  80°  to  66°. 
A9  —  80°  to  66°. 
ra  =  £  r,  to  4  TV 
A^  =  0.725  to  0.64. 
k  -0.75  to  1.75. 
P  —  4^-  inches  to  12  inches. 
/   =  ^,  =  same  as  for  axial-flow  turbines. 
Z,  =  Z  to  .  7  Z. 

(c)  For  radial  outward-flow  turbines,  where  Q  ranges 
from  2.5  to  350  cubic  feet  per  second,  and  h  ranges  from 
3  feet  to  25  feet, 

A   =  75°  to  66°  and  less. 
An_  =  80°  to  60°  and  less. 

KI  =  0.725  to  0.64. 
k   =  1.5  to  2. 


/    =  tl  =  same  as  for  axial-flow  turbines. 
Z,  ==  1.2  Z  to  1. 3  Z.     Art.  1 782. 

Velocity  of  Entrance. — From  the  general  relations 
select  a  value  of  A"t  to  correspond  with  the  type  of  wheel 
and  the  conditions  under  which  it  works;  then, 

(139.)     Art.  1783. 


130  TABLES   AND    FORMULAS. 

Effective  Area.  —  From  Q  and  ve  the  effective  area  A 
of  the  passages  from  the  guide  buckets  is  computed  from 
the  formula 

A=j-.  (140.)     Art.  1783. 

Radius.  —  From  this  value  of  A  the  mean  radius  of  a 
parallel-flow  wheel  is  computed  from  the  formula 

)     Art.  1783. 
where  k  is  a  coefficient  that  depends  on  the  relation  between 
<2  and  k.      (See  general  relations.) 

For  a  radial-flow  turbine,  the  radius  of  the  wheel  where 
the  water  enters  is  given  by  the  formula 

rl  =  k^A.  (1410.)     Art.  1783. 

where  k  depends  on  the  style  of  wheel,  whether  outward 
flow  or  inward  flow.      (See  general  relations.) 

Revolutions.  —  The  number  of  revolutions  per  minute 
is  given  by 


=  9.549  (142*.)     Art.  1783. 


for  axial-flow  turbines,  and  for  radial-flow  turbines, 
N=  9.549^'.  (1420.)     Art.  1783. 

Number  of  Vanes.  —  Having  chosen  the  pitch  P  approxi- 
mately to  suit  the  given  conditions,  the  number  of  guide 
vanes  for  an  axial-flow  turbine  is  given  by  the  formula 

Z=^,  (143«.)     Art.  1786. 

and  for  a  radial-flow  turbine  the  number  of  guide  vanes  is 
Z=^~^.  (1430.)     Art.  1786. 

These    formulas    give    approximate   values   for  Z,    and   the 
actual  value  is  the  nearest  corresponding  whole  number. 
The  number   of  wheel  vanes  Z,   for  axial-flow  turbines 


TABLES   AND    FORMULAS.  131 

should  always  be  greater  than  Z.      For  ordinary  cases  we 
may  take 

Z,=Z+2.  (144*.)     Art.  1787. 

For  radial  inward-flow  turbines  use  the  values 

Z1  =  Zto.7Z,  (1440.)     Art.  1787. 

and  for  radial  outward-flow  turbines 

Z,  =  1.2Zto  1.3Z.  (144f.       Art.  1787. 


Pitch.—  The  exact  pitch  for  the  guide  vanes  of  axial-flow 
wheels  is  now  given  by 

P=^Lt  (145«.)     Art.  1787. 

£ 

and  the  pitch  of  wheel  vanes  by 

Pl  =  "^.  (146*.)     Art.  1787. 

^j 

The  pitch  at  the  outflow  ends  of  guide  vanes  for  radial 
flow  turbines  is 

P  =  ^1.  (1450.)     Art.  1787. 

For  the  inflow  ends  of  the  wheel  vanes  the  pitch  is 
/^^p.  (1460.)     Art.  1787. 

"Width    of    Vanes.  —  Width    of    outflow    end    of    guide 
vanes, 


The  width  cl  of  the  inflow  end  of  the  wheel  vanes  is  made 
a  little  greater  than  e,  usually 

rE  =  ?  +  £  inch  to  f  +  f.  (148.)     Art.  1796. 

Width  of  outflow  end  of  wheel  vanes, 

ft  =  ^-^—.  (  1  49.)     Art.  1  796. 


132  TABLES   AND    FORMULAS. 

FORMULAS    USED   IN    HYDRAULIC   MACHINERY. 


SIZE   OF   AIR   AND   VACUUM    CHAMBERS. 

Let  V  =  volume  of  piston  displacement ; 
Vl=  volume  of  air  chamber; 
V^~  volume  of  vacuum  chamber. 

For  ordinary  double-acting  pumps  working  under  moder- 
ate pressures  at  ordinary  speeds, 

V^-^V.          Art.  1885. 

For  pressures  of  100  pounds  per  square  inch  and  upwards, 
or  for  high  piston  speeds, 

Vl  =  QV.          Art.  1885. 
For  ordinary  cases,  make 

F  =  iF.          Art.  1889. 


CALCULATIONS  RELATING   TO   PUMPS. 


Displacement. 

Let  D  =  displacement  in  cubic  feet  per  minute; 

d  =  diameter  of  piston  or  plunger  in  inches; 

L  •=.  length  of  stroke  of  piston  or  plunger  in  inches; 

N  —  number  of  discharge  strokes  made  by  piston  or 

plunger  in  1  minute. 
Then,       D  =  .000455  d*LN.  Art.  19O5. 

Slip. 

Let  s  =  slip ; 

D  =  displacement; 
C  —  actual  discharge. 

Then,  s=      ~     .  Art.  19O9. 

Head    and   Pressure. 

Let  H  —  head  in  feet ; 

P  =  pressure  in  pounds  per  square  inch. 

Art.  1914. 
Art.  1915o 


TABLES   AND   FORMULAS.  133 

Size    of   Piston    or    Plunger. 

Let  G  =  number  of  gallons  discharged  per  minute; 

5  =  speed  in  feet  per  minute  of  piston  or  plunger; 

d  =  diameter  of  piston  or  plunger  in  inches; 

F  =  number  of  cubic  feet  discharged  per  minute. 
Then,  the  theoretical  diameter  of  piston  or  plunger  is 

d  =  4.  95  V  ^  =  13.  54  |/^-.  Art.  1916. 

If  we  add  25  per  cent,  to  the  required  discharge  to  allow 
for  slip,  the  diameter  of  the  piston  or  plunger  will  be 

</=  5.535  V  ^  =  15.  138  V^-.  (152.)     Art.  1916. 

Discharge. 

The  theoretical  discharge  in  cubic  feet  equals  the  displace- 
ment. 

The  theoretical  discharge  in  gallons  per  minute  is 
.          Art.  1917. 


If  we  make  the  same  allowance  for  slip  as  was  made  in 
formula  152,  the  discharge  in  gallons  per  minute  is 
G  =.  03264  d*S;          (153.)     Art.  1917. 

and  in  cubic  feet  per  minute, 

F  =  .  00436  d*  S.  Art.  1917. 

Power. 

Let  H.  P.  =  horsepower; 

H  =  vertical  height  in  feet  from  the  surface  of  the 

water  in  the  well  or  sump  to  the  center  of  the 

outlet  end  of  discharge  pipe  ; 
G  =  discharge  in  gallons  per  minute; 
F  =  discharge  in  cubic  feet  per  minute. 

The  theoretical  power  is 

H.  P.  =  .000-254  G  H  =  .0019  F  H.  Art.  1918. 

If,    for    ordinary    cases,    the    frictional    resistances    are 


134  TABLES   AND    FORMULAS. 

assumed  to  be  50  per  cent,  of  the  power  developed  by  the 
engine,  the  power  required  is 

H.  P.  =  .00038  GH.  (154.)     Art.  1918. 

To  find  the  height  to  which  a  given  power  will  raise  a 
given  quantity  of  water,  making  the  same  allowance  for 
friction  as  in  the  last  formula, 


1919. 


Size  of  Steam  Cylinder. 

Let  S  =  steam  piston  speed  ; 

d  =  diameter  of  steam  cylinder  in  inches; 

r  =  ratio  between  length  of  stroke  and  diameter  of 
cylinder; 

/  =  length  of  stroke  in  feet; 

N  =  number  of  strokes  per  minute; 
H.  P.  .=  horsepower; 

P  =  steam  pressure  in  pounds  per  square  inch. 

Then,  for  simple  direct-acting  steam  pumps, 


rJ'N 


Art.  1920. 


Art 


i 

Having  obtained  the  diameter  of  the  steam,  piston  by 
either  of  the  above  formulas,  the  stroke  can  be  found  by 
multiplying  the  diameter  by  the  value  of  the  ratio  r.  When 
formula  157  is  used,  the  number  of  strokes  can  be  found 
by  dividing  the  piston  speed  by  the  length  of  the  stroke 
in  feet. 

Sizes  of  Suction  and  Delivery  Pipes. 

For  a  velocity  of  200  feet  per  minute  in  the  suction  pipe 
and  400  feet  per  minute  in  the  delivery  pipe, 

Let  d^  =  diameter  of  suction  pipe; 
d^  =  diameter  of  delivery  pipe; 
G  —  discharge  in  gallons  per  minute. 


TABLES    AND    FORMULAS.  135 


=  4.95  Y~,  or  dl  =  .35  \/G.  (158.)     Art.  1921. 


<  =  4.95  f  —  ,  or  <f9  =  .25  \/G.  (159.)     Art.  1921. 

The  pipes  may  be  made  larger  than  the  values  calculated 
by  the  above  formulas,  particularly  the  suction  pipe,  but  it 
is  not  good  practice  to  make  them  any  smaller. 

DUTY    OF    A    PUMP. 

Old  Standard.  —  According  to  the  old  standard,  the  duty 

of  a  pumping  engine  is  the  number  of  pounds  of  ivater  raised 

one  foot  high  for  each  100  pounds  of  coal  burned  in  the  boiler. 

Let   G  =  number  of  gallons  discharged  in  a  given  period  • 

h  =  total  vertical  distance  in  feet  from  the  surface 

of   the   water    in    the   well,    or  other   source  of 

supply,  to  the  point  of  discharge; 

W  =  the   number  of  pounds    of  coal  burned  in  the 

given  period  ; 

D  =  the  duty  in  foot-pounds. 
Then, 


Standard   Recommended  by  Committee   of  Ameri- 
can Society  of  Mechanical  Engineers. 

The  duty  of  a  pumping  engine  is  equal  to  the  total 
number  of  foot-pounds  of  work  actually  done  by  the  pump, 
divided  by  the  total  number  of  heat  units  in  the  steam  used 
by  the  pump,  including  the  steam  used  by  the  condensers 
(if  any)  and  boiler  feed,  and  this  quotient  multiplied  by 
1,000,000. 

The  number  of  foot-pounds  of  work  done  by  the  pump  is 
to  be  found  as  follows  :  A  pressure  gauge  is  attached  to  the 
discharge  pipe  and  a  vacuum  gauge  to  the  suction  pipe, 
both  as  near  the  pump  as  convenient;  then  the  pressure 
against  which  the  pump  plunger  works  is  equal  to  the 
difference  in  the  pressures  shown  by  these  two  gauges  plus 


136  TABLES   AND    FORMULAS. 

the  head  due  to  the  difference  in  level  of  the  points  in  the 
pipes  to  which  they  are  attached;  and  the  number  of  foot- 
pounds is  equal  to  the  continued  product  of  the  net  area  of 
the  plunger  (making  allowance  for  piston  rods),  the  length 
of  the  plunger  stroke  in  feet,  the  number  of  plunger  strokes 
made  during  the  trial,  and  the  pressure  against  which  the 
pump  plungers  work,  as  shown  by  the  gauges. 

The  number  of  heat  units  furnished  to  the  pump  is  the 
number  of  British  thermal  units  (B.  T.  U.)  in  the  steam 
from  the  boilers,  and  is  to  be  determined  by  an  evaporation 
test  of  the  boilers.  If  we  let 

A  =  the  net  area  of  the  plunger  in  square  inches; 

P  =  the  pressure  in  pounds  per  square  inch  indicated  by 

the  gauge  on  the  discharge  pipe  ; 

p  =  the  pressure  in  pounds  per  square  inch  corresponding 
to  the  vacuum  indicated  by  the  gauge  on  the  suction 
pipe; 
5  =  the  pressure  in  pounds  per  square  inch  corresponding 

to  the  difference  in  level  between  the  two  gauges; 
L  =  the  average  length  of  stroke  of  pump  plunger  in  feet  ; 
N  —  the  total  number  of  single  strokes  of  plunger  made 

during  the  trial; 
H  =  the  total  number   of   heat  units  consumed    by  the 

engine  during  the  trial; 
W=  the  total  number  of  foot-pounds  of  work  done  by  the 

pump  during  the  trial;  and 
D  =  the  duty. 
Then,W=A(P±/>  +  S)LN,  (161.)     Art.  1924. 


W 
and  Z>  =  -,  x  1,000,000  = 


(162.)     Art.  1924. 


CALCULATIONS    RELATING    TO    HYDRAULIC    MACHINERY. 


Relations  Between  Pressure  and  Size  of  Ram. 

Let  D  =  the  diameter  of  a  hydraulic  piston  or  ram ; 

W  •=•  the  weight   of   the  ram  and  attachments  that 
must  be  lifted  by  the  water; 


TABLES   AND    FORMULAS.  137 

p  =  the  pressure  of  the  water  in  pounds  per  square 

inch; 

/'"=  the  percentage  of  friction; 
/-*  =  the  net  pressure  exerted  by  the  ram. 
To  find  the  net  pressure  exerted  by  a  ram  or  plunger  of  a 
hydraulic  press, 

P=  .7854  X  £>2X/>X  (I-T£)  -  W.  (163.) 

Art   1969. 

To  find  the  pressure  per  square  inch  required  to  exert  a 
given  net  pressure  when  the  diameter  and  weight  of  the 
ram  and  the  percentage  of  friction  are  given,  use  the  for- 
mula 

p\    w 
/=-  —^r.  (164.)     Art.  1970. 


To  find  the  diameter  of  piston  or  ram  required  to  exert 
a  given  net  pressure,  use  the  formula 


/>==/-  (165.)     Art.  1971. 


"Weight   and   Volume   of  Accumulators. 

Let  IT,  =  Aveight  of  accumulator  ram; 

]l'\  —  load  on  accumulator  ram; 

/},  =  diameter  of  accumulator  ram ; 

/,  =  maximum  pressure  per  square  inch  in  the  accu- 
mulator cylinder; 

ft  =  minimum  pressure  per  square  inch  in  the  accu- 
mulator cylinder; 

p    =  mean  pressure  per  square  inch  in  the  accumu- 
lator cylinder; 

5   =  stroke  of  accumulator  ram ; 

V  =  total  volume  of  water  displaced  by  accumulator 
ram  during  the  stroke  S; 

F  =  the  percentage  of  friction. 

To  find  the  mean   pressure  /  corresponding  to  a  given 
case,  use  the  formula 


138  TABLES   AND   FORMULAS. 


The  maximum  pressure  is  found  by  the  formula 

W  4-  W 
?,=  —r-  (167.)     Art.  1973. 


and  the  minimum  pressure  by 

(168.)     Art.  1973. 


The  "weight  required  to  produce  a  given  mean  pressure 
when  diameter  and  weight  of  the  ram  are  known  may  be 
found  from  the  following  formula: 

fF,  =  ,7854x/V  X/-  W;.  (169.)     Art.  1974. 

The  relations   between  the  stroke,  diameter,  and  volume 
of  an  accumulator  are  given  by  the  following  formulas: 
V  =  .7854  /VS.  (17O.)     Art.  1977. 


777 
Z\  =  1.128  f-^.  (172.)     Art.  1977. 

o 

In  the  above  formulas,  if  7)l  and  vS  are  in  inches,  the 
volume  will  be  given  in  cubic  inches;  and  if  Dl  and  S  are  in 
feet,  V  will  be  given  in  cubic  feet. 


FORMULAS    USED    IN    WATER    SUPPLY    AND 
DISTRIBUTION. 


DIMENSIONS  OF   SPILLWAY   OR   O%7ERFLOW. 

Let  Z=  length  of  lip  of  spillway  in  feet; 

A  —  area  of  watershed  above  dam  in  square  miles; 
D  —  depth  of  notch  of  spillway  in  feet ; 
Q  •=.  cubic  feet  of  water  per  second  per  square  mile; 
C  =  a  constant   depending  on  the    character  of    the 

dam  and  its  surroundings  and  the  area  of  the 

watershed. 


TABLES   AND    FORMULAS.  139 

Then,      L  =  W^A.  (173.)     Art.  2O48. 


C.  (174.)     Art.  2048. 


16 

If  we  assume  Q  =  64,  which  corresponds  to  a  little  over 
41  million  gallons  per  24  hours,  per  square  mile,  and  repre- 
sents a  very  powerful  freshet  flow,  although,  perhaps,  not 
the  maximum,  formula  174  reduces  to 

D  =  \/~A  +  C.  (175.)     Art.  2O49. 


MASONRY    DAMS. 

Let     A    =  thickness  of  top  of  a  trapezoidal  dam  in  feet ; 
B    =  thickness  of  base  of  dam  in  feet ; 
C    =  a   factor  of   safety  against    either    sliding   or 

overturning; 
D    —  density  (weight  per  cubic  foot)  of  material  of 

which  dam  is  built ; 

H  —  head  of  water  pressing  against  the  dam  in  feet ; 
R    =  resistance  of  wall  to  sliding; 
T   =  horizontal  thrust  in  pounds  on  the  dam,  due  to 

the  head//"; 

MR  =  moment   of   resistance  of   dam   against  over- 
turning by  rotating  about  its  outer  toe; 
M  T  —  moment  of  thrust  about  the  outer  toe  of  the 

dam. 
The  thrust  is 

r=31.25#a,  (176.)     Art.  2063. 

and  the  moment  of  thrust 

MT=10A2H\  (177.)     Art.  2O63. 

The  resistance  of  the  wall  to  sliding  is 

R  =  0.1! 5 AD.  (178.)     Art.  2O65. 

The  moment  of  resistance  to  overturning  for  a  wall  with 
vertical  sides  is 

7)  //"  7'2 

-      — ,  (179.)     Art.  2066. 


140  TABLES  AND    FORMULAS. 

and  for  a  trapezoidal  wall 


MR  =  ^(A  B  -  ~-  +  /A  (  1  80.)     Art.  2O66. 

The    relation    between    A,  B,  D,  and  H  for    a    factor   of 
safety  C  against  sliding  is  given  by  the  formula 

B  =  ^-™DCH  -A.  (181.)     Art.  2067. 

For  a  factor  of  safety  C  against  overturning,  the  breadth 
of  the  base  is  given  by  the  formula 


£  =  $  +  3A*--  (182.)     Art.  2068. 

JJ  A 

Average  Dimensions.  —  For  practical  values  of  A  and 
,  a  satisfactory  value  of  B  is 

B  =  f  H  to  \H.          Art.  2O7O. 


HIGH   MASONRY   DAMS. 

Maximum    Unit    Stress    on    Base    of   Dam    for 
Unequally    Distributed    Load. 

Let      L     =  length  of  base  of  a  section  through  the  dam; 

d     =  length  of  the  shorter  segment  of  this  base; 
L  —  d—  length  of  the  remaining  segment; 
W    =  the  resultant  of  the  weight  of  the  section,  or 

the  vertical  component  of  this  resultant; 
P     —  maximum  unit  stress. 

There  are  three  empirical  formulas  for  the  value  of  Pt 
which  experience  shows  give  satisfactory  results;  viz. : 

P=^f(L-\.Zd),     (183.)  Art.  2072. 

o  w 
P=~j,     (1 84.)  Art.  2072. 

and    P  =  K(^~^.     (185.)  Art.  2O72. 

Of  these  formulas,  the  last   is   probably   the   most  satis- 
factory. 


TABLES  AND    FORMULAS.  141 

DARCY'S     FORMULAS. 

Let  D  =  diameter  of  pipe  in  feet ; 
H  —  total  head  in  feet; 
L  =  total  length  in  feet; 
V=  velocity  of  efflux  in  feet  per  second; 
C  =  an  experimental  coefficient  (see  table  of  Coeffi 

cients  for  Darcy's  Formula) ; 

Q  =  quantity  discharged  in  cubic  feet  per  second; 
A  =  area  of  pipe  in  square  feet ; 

Ti- 
ll —  head  per  1,000  feet  of  length  =  —     -—=-. 

I  ?  000  -Z-- 

Fundamental  Formulas  for  Long  Pipes. 

1.  (186.)     Art.  2O92. 


(187.)     Art.  2092. 


Q--Air£.          (188.)     Art.  2092. 

/75~Tr 

Q  =  0.7854  D*V±L±L.  (189.)     Art.  2O92. 


(190.)     Art   2092. 


Approximate  Formulas  for  Rough  Pipes 

For  pipes  from  8  inches  to  48  inches  in  diameter, 


Art.2094. 

(192.)     Art.  2O94. 
(193.)     Art.  2094. 
(  1  94.)     Art.  2094. 
Art.2094. 


142  TABLES  AND   FORMULAS. 

For  pipes  from  3  inches  to  6  inches  in  diameter, 
j2_  =  0.785.  (196.)     Art.  2O94. 

0=0.894/^1.  (197.)     Art.  2094. 

Formulas  for  Smooth  Pipes. 

Q=^%D'/i.     (198.)  Art.  2O95. 

£7)5=2.    (199.)  Art.  2O95. 
(2=1.404/271    (2OO.)  Art.  2O95. 

General    Relation    Between    Smooth    and    Rough 
Pipes. 

In  general,  the  discharge  tJirougJi  a  smooth  pipe  is  1.40 
times  t/iat  through  a  rough  pipe  of  the  same  diameter;  and, 
reciprocally,  the  discharge  through  a  rough  pipe  is  0. 70  times 
that  through  a  smooth  one  of  the  same  diameter.  These 
factors  represent  the  practical  limits  between  which  the 
extremes  of  smoothness  and  roughness  can  affect  the  flow. 
Art.  2O95. 

Formulas  for  Velocity. 

For  rough  pipes  of  from  8  inches  to  48  inches  in  diameter, 

F=1.274/Z>1.  (201.)  Art.  2096. 

For  rough  pipes  of  smaller  diameter, 

F=1.134/2>1  (2O2.)  Art.  2O96. 

For  smooth  pipes  of  large  diameter, 

V=  1.78  4/2?l.  (2O3.)  Art.  2O96. 

For  smooth  pipes  of  small  diameter, 

F=  1.60  4/2)1.  (2O4.)     Art.  2O96. 


TABLES  AND   FORMULAS.  143 

General  Relation   Between    the   Elements   of   Two 
Pipes. 

Let  D,  Q,  L,  //,  and  C  be  the  respective  elements  of  one 
pipe  and  77,  Q',  L ',  //',  and  C'  the  similar  elements  of 
another;  then, 

DHC'L'V* 


D'H'CL  V 
If,  as  can  usually  be  done,  we  make  C  —  C',  we  have 

*  =  L  (205.)     Art.  2O97. 


Also,  =  I-  (206.)     Art.  2O97. 


If  L  and  H  equal,  respectively,  L'  and  //', 

Tj  =  T  ^T-  (207.)     Art.  2O98. 

To  find  the  number  x  of  small  pipes  with  the  diameter  D' 
to  replace  a  pipe  whose  diameter  is  D, 

x  =  fX-1.          (208.)     Art.  2098. 


COMPOUND   PIPES. 

To  find  the  diameter  of  a  simple  pipe  that  will  give  the 
same  delivery  as  a  given  compound  system: 

Let  D  =  diameter  of  the  simple  pipe; 
L  —  length  of  the  simple  pipe; 
d,  d' ,  d" ,  etc.  =  diameters  of  the   respective  sections 

of  the  compound  pipe; 
/,  /',  /",  etc.  =  lengths  of  the  respective  sections  of 

the  compound  pipe. 

Then, 

•  =    -• +    *  +    "* +  etc'         (209°    Art'  21 1 1' 


144  TABLES  AND    FORMULAS. 

PUMPING   INTO  MAINS. 

Theoretical  horsepower  required  to  force  a  given  quantity 
of  water  into  a  main  against  a  given  pressure  head: 

Let  H.  P.  =  theoretical  horsepower; 
H  —  pressure  head  in  feet; 
<2  =  quantity  of  water  in  cubic  feet  per  second. 


H.  P.  =  '.  (210.)     Art.  2117. 

0.0 


"WEIGHTS  ANI>  THICKNESS  OF  CAST-IRON   PIPES. 

Let  W  —  weight  in  pounds; 
D   —  diameter  in  inches; 
T   ~  thickness  in  inches; 
L    —  length  in  inches; 
P   =  weight  in  long  tons  (2,240  pounds); 
M   =  length  in  miles  ; 

W  =  approximate  weight  per  foot  in  pounds; 
H  =  total  head  in  feet. 


T)  Tx  L.  (211.)     Art.  2125. 

T}  T.  (212.)     Art.  2126. 

P=Z5Jlf(D  +  T)  T.  (213.)     Art.  2127. 

T  =  0.  00006  HD+  0.01  337?  +  0.290.     (214.)  Art.  2128. 


DARCY'S  FORMULAS   FOR   FLOW   IN   OPEN  CHANNELS. 

Let      U  =  mean  velocity  of  flow  in  feet  per  second ; 

5  —  water  section  in  square  feet; 
W  P—  wet  perimeter  in  feet; 

R  =  mean  hydraulic  radius  =  -rrj-n't 

I  =  slope  of  free  water  surface  per  foot  of  length  = 

total  fall  of  surface  divided  by  total  length ; 
D  =  interior  diameter  of  a  circular  conduit  in  feet. 


TABLES  AND    FORMULAS.  145 

For  an  ordinary   tunnel  or  channel   lined  with  well-laid 
brick. 


Art.  2,43. 


For  a  circular  brick-lined  conduit  running  full, 


FORMULAS  USED   I1V  IRRIGATION. 


APPROXIMATE  DISCHARGE   OF   WEIRS. 

Let    /  =  length  of  notch  in  feet; 

//=  measured  head  on  crest  in  feet; 
Q  —  discharge  in  cubic  feet  per  second; 

then,         Q=^IH\  (217.)     Art.  2163. 


FLOW   OF   WATER   THROUGH   CONDUITS. 

Let  //  =  difference  in  level  between  the  ends  of  the  canal, 

or  any  two  cross-sections  of  the  canal ; 
/  =  horizontal   length   of   that  portion   of  the  canal 
included  between  the  sections  whose  difference  of 
level  is  // ; 

s  =  slope  =  the  ratio  y; 

a  —  area  of  the  water  cross-section; 
p  =  wetted  perimeter; 

r  =  hydraulic  radius  —  the  ratio  — ; 

c '  =  a  coefficient  depending  on  the  nature  ot  the  sur- 
face of  the  conduit; 
and      i>  =  mean  velocity  of  flow. 

The  laws  for  the  resistance  to  flow  may  be  expressed  by 
the  relation  Ji  a  =  c'  I  p  i? ,  from  which  we  have  the  general 
formula 


v  =  <-  X  a-  =  \~  xsxr.          (218.)     Art.  21 73. 


140  TABLES   AND    FORMULAS. 

By    replacing  \  —,  by    the    equivalent   factor  r,   we   have 
v  —  c\/rs,  the  same  as  formula  5O,  Art.  1O33. 
Formulas  for  Flow  in  Canals. 

Canals  with  earthen  banks, 


Canals  lined  with  dry  stone, 

(22O.)     Art.  2183. 


Canals  lined  with  rubble  masonry, 

v  —  r~s-^ — 'iTT7"-  (221.)     Art.  2184. 

Wooden  flumes, 


TIMBER  FOR   PLUMBS. 

Let  W=  total  load  in  pounds  carried  by  any  beam; 
/  =  length  of  beam  in  inches; 
b  =  breadth  of  beam  in  inches; 
d  =  depth  of  beam  in  inches; 

S  —  maximum  unit  fiber  stress  in  pounds  per  square 
inch. 

For  a  simple  beam  with  a  uniformly  distributed  load, 
pr-fi^S.  (222.)    Art.  2189. 

For  a  simple  beam  with  a  concentrated  load  at  the  middle, 
W=%b--S.  (223.)     Art.  2190. 


TABLES   AND    FORMULAS. 


147 


For  a  beam  with  a  concentrated  load  at  a  distance  /:  from 
one  support  and  /„  from  the  other,  where  /:  -{-  /2  =  /, 


(224.)     Art.  2191. 


l.L 


For  a  beam  on  which  the   load   at  one  end   is  zero,  with  a 
uniform  increase  in  the  load  to  the  other  end, 

W=1.3~-S.  (225.)     Art.  2192. 


SAFE   WORKING   STRESS  S. 

For  good  sound  timber. 


Kind  of  Timber. 

Safe  Working  Stress. 

Steady  Load. 

Variable  Load. 

Yellow  Pine  
White  Oak  

i,  800 
1,350 
1,250 
1,200 

1,100 

1,200 
1,000 
900 

800 

Spruce 

Hemlock  

White  Pine 

TRUSSES. 

Trussed  Stringers. 

Let    L  =  span  in  inches; 

H  —  depth  of  truss  in  inches; 

b  =  breadth  of  stringer  in  inches; 
d  =  depth  of  stringer  in  inches; 
Wt  =  total  uniformly  distributed  load  in  pounds; 
5  =  allowable  unit  fiber  stress  in  stringer  or  strut; 
S^  —  total  stress  in  tie-rods; 

h  —  width  of  strut  in  inches; 

t   =  thickness  of  strut  in  inches. 


148  TABLES   AND    FORMULAS. 

Relation  between  Wt  and  dimensions  of  stringer, 


Stress  in  tie-rods, 

Art.  2198. 


Stress  in  strut, 

Wt  =  -\htS.  (229.)     Art.  2198. 

The  King-Rod   Truss. 

Let   L  —  length  of  span  in  inches; 

Wt=  total  uniformly  distributed  load  in  pounds; 
W  =  total  stress  in  each  strut  in  pounds; 
Ss  =  safe  unit  stress  in  king-rods; 
A   =  net  sectional  area  of  king-rods; 
and       H  —  depth  of  truss  in  inches^; 


(230.)     Art.  2199. 
and  A=$^-.  (231.)     Art.  2199. 

The  Queen-Rod  Truss. 

Let  St  —  maximum  unit  stress  in  tie-beam; 

W  =  total  uniformly  distributed  load  in  pounds; 
L  —  length  of  span  in  inches; 
H  =  depth  of  span  in  inches; 
b  =  breadth  of  tie-beam  in  inches; 
d  =  depth  of  tie-beam  in  inches; 
Sc  =  total  stress  in  upper  chord  member  in  pounds; 
5S  =  total  stress  in  struts  in  pounds; 
Sq  =  allowable  unit  stress  in  queen-rods; 
A  =  sectional  area  of  queen-rods. 

The  maximum  unit  stress  in  tie-beam  is 

(232.)     Art.  22OO. 


TABLES  AND  FORMULAS.  N'.t 

The  total  stress  in  the  upper  chord  member  is 
Sc  =  %^.  (233.)     Art.  22OO. 

The  total  stress  in  each  strut  is 

* 

S.=  nVi  +  -6V^-.  (234.)     Art.  2200. 

The  net  area  of  each  queen-rod  is 

A=^^-.  (235.)     Art.  22OO. 

•^fl 

The  Howe  Truss. 

Let  Nt  =  the  number  of   a  tie,  counting  from  the  center; 
St  =  total  stress  in  a  tie ; 
L,  =  length  of  a  tie  ; 
Ls  =  length  of  a  strut; 
5S  =  total  stress  in  a  strut ; 
A^   =:  number  of  panels  in  truss  from  center  to  either 

abutment; 
n    =  number  of   panels  from  a   given  panel    to  the 

nearer  abutment  ; 
P   =  panel  load  in  pounds  ; 
Lp  =  length  of  a  panel  ; 

St<.  =  total  stress  in  top  chord  of  a  given  panel  ; 
Sb(.  =  total  stress  in  bottom  chord  of  a  given  panel. 

The  stress  in  any  tie  whose  number  is  Nt  is 

S,  =  (iVt  +  i)  P.  (236.)     Art.  22O2. 

The  stress  in  a  strut  is 

5S  =  *pSt.  (237.)     Art.  22O2. 

**t 

The  top  chord  stress  is 

Ste  =  nP(N-$  n}  ^.  (238.)     Art.  22O2. 


150  TABLES   AND    FORMULAS. 

The  bottom  chord  stress  is 


REFUSAL  OF   PILES. 

Let  5  =  weight  a  pile  will  bear  with  safety; 

W  —  weight  of  hammer,  in  the  same  unit  as  S; 
H  —  height  of  fall  of  hammer  in  feet; 

then,  5  =  W H.  (24O.)     Art.  22O5. 


INDEX. 


TABLES.  PAGE 

Common  Logarithms         .        .  1-19 

Natural  Sines,  Cosines,  Tangents, 

and  Cotangents  .  .  .  21-40 
Traverse  Tables,  or  Latitudes  and 

Departures  of  Courses  .        .          41-49 
Horizontal   Distances   and    Differ- 
ences of  Elevation    for  Stadia 
Measurements          .        .        .          51-60 
Radii  and  Chord  and  Tangent  De- 
flections              61-63 

Moments  of  Inertia  ....  64 
Bending  Moments  and  Deflections  65 
Specific  Gravities  and  Weights  per 

Cubic  Foot        ....          66-68 
Discharge  of  Standard  Orifices         68-69 
"  "   Weirs    ....      70 

Coefficients  of  Friction  for  Smooth 
Cast  or  Wrought  Iron 
Pipes    .... 
for  Angular  Bends 
"  "   Circular  Bends 

"  "    Darcy's  Formula  . 

Properties  of  Saturated  Steam         73- 
Standard  Dimensions  of  Wrought- 
Iron    Steam,   Gas,   and 
Water  Pipes  ...      76 
"         Pipe  Flanges     ...       77 
Specific  Heat  of  Substances    .        .       78 
Constants   for    Apparent    Cut-Offs 

Used  in  Determining  M.  E.  P.  .  78 
Riveted  Joints  of  Boilers  ...  78 
Positions  of  Eccentric  Relative  to 

Crank 79 

Diameters  of  Steam  and  Exhaust 

Pipes 79 

Piston  Speeds  of  Steam  Engines  .  79 
Ratio  of  Grate  Area  of  Boiler  to 

Horsepower     ...      79 
"      "    Heating  Surface  to  Grate 

Area  ....      80 

"      "    Heating  Surface  to  Horse- 
power       ....      80 
Factors  of  Evaporation     .        .        .      81 
Size  of  Chimneys  and  Horsepower 
of  Boilers  .        .      82 


RULES  AND  FORMULAS.  PAGE 

Formulas  Used  in  Algebra       .        .  83 

Trigonometric  Functions          .        .  83 
Rules  for  Using  Tables  of    Loga- 
rithms of  Num- 
bers .        .          84-86 
"        "         "         Trigonometric 

Tables      .        .  86 

RULES  FOR  MENSURATION. 

The  Triangle 87 

'*    Quadrilateral  ....  87 

"    Circle 87 

"    Ellipse 88 

"    Prism  and  Cylinder       .        .  88 

"    Pyramid  and  Cone         .        .  89 
"    Frustum  of    a    Pyramid    or 

Cone 89 

"    Sphere 89 

FORMULAS  USED  ix  ELEMENTARY 

MECHANICS. 

Uniform  Motion       ....  89 

Mass,  Weight,  and  Gravity  .        .  90 

Formulas  for  Gravity  Problems  90 

Falling  Bodies         ....  90 

Centrifugal  Force   ....  91 

Center  of  Gravity  of  Two  Bodies  92 

The  Efficiency  of  a  Machine          .  92 

Work         ...                 .        .  92 

Power 92 

Kinetic  Energy        .        .        .        -92 

Density 93 

RULES   AND  FORMULAS   USED   IN 
HYDRAULICS. 

Pascal's  Law 93 

General  Law  for  the  Downward 
Pressure  Upon  the    Bottom    of 

Any  Vessel 93 

General  Law  for  Upward   Pres- 
sure           93 

General  Law  for  Lateral  Pressure  94 

"           "       "   Pressure     .        .  94 

Specific  Gravity       ....  94 

Mean  Velocity          ....  95 

Velocity  of  Efflux  from  an  Orifice  95 


RULES  AND  FORMULAS   USED    IN 

HYDRAULICS— Continued.          PAGE 
Theoretical  Range  of  a  Jet    .        .      96 
Velocity  of  a  Jet      .        .        .        .96 
Discharge  of  an  Orifice  ...      97 
"  "  Standard  Orifices    .      97 

"  "  a    Submerged  Rect- 

angular Orifice     .      98 
"  Weirs         ...      98 
Flow  of  Water  Through  Pipes     99-102 
"  Water  Through  Conduits 

and  Channels          .      102-104 
Values  of  the  Coefficient  of  Rough- 
ness  for  Use  in   Kutter's   For- 
mula        103 

FORMULAS  USED  IN  PNEUMATICS. 
Pressure,  Volume,  Density,  and 
Weight  of  Air  When  the  Tem- 
perature Is  Constant  .        .        .104 
Mariotte's  Law        ....     104 
Pressure  and   Volume  of  a  Gas 

with  Variable  Temperature  .  105 
Gay-Lussac's  Law  .  .  .  .105 
Mixture  of  Two  Gases  Having 

Unequal  Volumes  and  Pressures   106 
Mixture  of  Two  Volumes  of  Air 
Having  Unequal  Pressures,  Vol- 
umes, and  Temperatures   .        .     106 

FORMULAS  USED  IN  STRENGTH  OF 

MATERIALS. 

Unit  Stress,  Unit  Strain,  and  Co- 
efficient of  Elasticity  .        .        .106 
Strength  of  Pipes  and  Cylinders       107 
Moment  of  Inertia,  Resisting  Mo- 
ment, and  Bending  Moment  of 

Beams 107 

Deflection  of  a  Beam  .  .  .  108 
Strength  of  Columns  .  .  .  108 
' '  "  Shafts  ....  109 
Constants  for  Shafting  .  .  .no 
Strength  of  Ropes  and  Chains  .  no 

FORMULAS  USED  IN  SURVEYING. 

Radius  of  a  Curve  .        .        .  .in 

Length  of  Subchords      .        .  .in 

Length  of  a  Tangent  of  a  Curve  .     in 

Chord  Deflection     .        .        .  .112 

Tangent  Deflection         .        .  .     112 

Stadia  Measurements    .        .  .112 

Barometrical  Leveling  .        .  .     113 

RULES  AND   FORMULAS  USED  IN 
SURVEYING  AND  MAPPING. 

Rule  for  Balancing  a  Survey        .     113 
"       "    Double  Longitudes         .     113 

Application     of    Double    Longi- 
tudes to  Finding  Areas       .        .     114 


RULES  AND    FORMULAS   USED  IN 
SURVEYING     AND     MAPPING— 

Continued.  PAGE 

Trapezoidal  Rule    .        .        .  .114 

Simpson's  Rule        .        .        .  .114 

Volumes  of  Irregular  Solids  .     115 

The  Prismoidal  Formula       .  .     115 

Latitudes  and  Departures     .  .     115 

FORMULAS   USED  IN    STEAM   AND 
STEAM  ENGINES. 

Specific  Heat 115 

Temperature  of  Mixtures      .         .  116 

Mixture  of  Steam  and  Water        .  116 
Work  Done  by  Piston    .        .        .116 

Real  and  Apparent  Cut-Off  .        .  116 

Horsepower 117 

Mean  Effective  Pressure        .        .  117 

Piston  Speed n7 

Mechanical  Efficiency  of  Engine 
Steam  Consumption 
Thermal  Efficiency  of  Engine 

Water  Required  by  Condenser     .  n8 
Ratio  of  Expansion         .        .        .118 

FORMULAS  USED  IN  STEAM  BOIL- 
ERS. 
Air    "Required    for    Combustion 

and  Heat  of  Combustion  .  .  119 
Strength  of  Boiler  Shells  .  .119 
Horsepower  of  Boilers  .  .  .  119 

Safety  Valve J2o 

Draft  Pressure  of  Chimney  .        .     120 
Quality  of  Steam    .        .        .        .121 

FORMULAS  USED  IN  WATER-WHEELS. 

Theoretical  Energy  of  a  Given 
Head  and  Weight  of  Water  .  121 

Theoretical  Power          .        .        .121 

Energy  of  a  Jet       .        .        .        .     122 

Pressure  Due  to  Impact  and  Re- 
action of  a  Jet 

Efficiency 

Overshot  Water- Wheels 

Breast  Wheels 

Undershot  Wheels  . 

Poncelet's  Wheel    . 

Turbines  . 


.  122 

.  124 

.  124 

•  125 
.  125 

•  125 
126-131 

FORMULAS  USED  IN  HYDRAULIC 
MACHINERY. 

Size  of  Air  and  Vacuum  Cham- 
bers   

Displacement  of  Pumps 


Slip 

Head  and  Pressure 

Size  of  Pump  Piston  or  Plunger  . 

Discharge  of  Pumps 

Power  of  Pumps     .... 


FORMULAS   USED   IN    HYDRAULIC 

MACHIX  ERY— Con  finued.  PAGE 

Size  of  Steam  Cylinder  for  Pumps    134 
Sizes    of    Suction    and    Delivery 

Pipes 134 

Duty  of  a  Pump       .        .        .        .135 
Relations  Between  Pressure  and 

Size  of  a  Ram       .        .        .        .136 
Weight  and  Volume  of  Accumu- 
lators 


FORMULAS  USED  IN  WATER  SUP- 
PLY AND  DISTRIBUTION. 
Dimensions  of  Spillway  or  Over- 
flow         

Masonry  Dams        .... 
High  Masonry  Dams      . 
Darcy's  Formulas  for  Long  Pipes 
"  "  "  Rough  Pipes 

"  "  Smooth  Pipes 

General     Relations      Between 
Smooth  and  Rough  Pipes 


37 


FORMULAS  USED  IN  WATER  SUP- 
PLY AND  DISTRIBUTION— Cent.  PAGE 

Darcy's  Formulas  for  Velocity     .     142 

General  Relation  Between  Ele- 
ments of  Two  Pipes  . 

Compound  Pipes     .... 

Pumping  Into  Mains 

Weights  and  Thickness  of  Cast- 
iron  Pipes  

Darcy's  Formulas  for  Flow  in 
Open  Channels  .... 

FORMULAS  USED  IN  IRRIGATION. 
Approximate  Discharge  of  Weirs 
Flow  of  Water  Through  Conduits 
"      in  Canals         .... 
Timber  for  Flumes 
Trussed  Stringers  .... 
King-Rod  Truss      .... 
Queen-Rod  Truss    .... 

Howe  Truss 

Refusal  of  Piles 


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